First perfect square in base n with n unique digits
Find the first perfect square in a given base N that has at least N digits and exactly N significant unique digits when expressed in base N.
E.G. In base 10, the first perfect square with at least 10 unique digits is 1026753849 (32043²).
You may use analytical methods to reduce the search space, but the code must do a search. Do not use magic numbers or just feed the code the answer to verify it is correct.
- Task
- Find and display here, on this page, the first perfect square in base N, with N significant unique digits when expressed in base N, for each of base 2 through 12. Display each number in the base N for which it was calculated.
- (optional) Do the same for bases 13 through 16.
- (stretch goal) Continue on for bases 17 - ?? (Big Integer math)
- related task
C++
A stripped down version of the C#, using unsigned longs instead of BigIntegers, and shifted bits instead of a HashSet accumulator. <lang Cpp>#include <string>
- include <iostream>
- include <cstdlib>
- include <math.h>
- include <chrono>
- include <iomanip>
using namespace std;
const int maxBase = 16; // maximum base tabulated int base, bmo, tc; // globals: base, base minus one, test count const string chars = "0123456789ABCDEF"; // characters to use for the different bases unsigned long long full; // for allIn() testing
// converts base 10 to string representation of the current base string toStr(const unsigned long long ull) { unsigned long long u = ull; string res = ""; while (u > 0) { lldiv_t result1 = lldiv(u, base); res = chars[(int)result1.rem] + res; u = (unsigned long long)result1.quot; } return res; }
// converts string to base 10 unsigned long long to10(string s) { unsigned long long res = 0; for (char i : s) res = res * base + chars.find(i); return res; }
// determines whether all characters are present bool allIn(const unsigned long long ull) { unsigned long long u, found; u = ull; found = 0; while (u > 0) { lldiv_t result1 = lldiv(u, base); found |= (unsigned long long)1 << result1.rem; u = result1.quot; } return found == full; }
// returns the minimum value string, optionally inserting extra digit string fixup(int n) { string res = chars.substr(0, base); if (n > 0) res = res.insert(n, chars.substr(n, 1)); return "10" + res.substr(2); }
// perform the calculations for one base void doOne() { bmo = base - 1; tc = 0; unsigned long long sq, rt, dn, d; int id = 0, dr = (base & 1) == 1 ? base >> 1 : 0, inc = 1, sdr[maxBase] = { 0 }; full = ((unsigned long long)1 << base) - 1; int rc = 0; for (int i = 0; i < bmo; i++) { sdr[i] = (i * i) % bmo; if (sdr[i] == dr) rc++; if (sdr[i] == 0) sdr[i] += bmo; } if (dr > 0) { id = base; for (int i = 1; i <= dr; i++) if (sdr[i] >= dr) if (id > sdr[i]) id = sdr[i]; id -= dr; } sq = to10(fixup(id)); rt = (unsigned long long)sqrt(sq) + 1; sq = rt * rt; dn = (rt << 1) + 1; d = 1; if (base > 3 && rc > 0) { while (sq % bmo != dr) { rt += 1; sq += dn; dn += 2; } // alligns sq to dr inc = bmo / rc; if (inc > 1) { dn += rt * (inc - 2) - 1; d = inc * inc; } dn += dn + d; } d <<= 1; do { if (allIn(sq)) break; sq += dn; dn += d; tc++; } while (true); rt += tc * inc; cout << setw(4) << base << setw(3) << inc << " " << setw(2) << (id > 0 ? chars.substr(id, 1) : " ") << setw(10) << toStr(rt) << " " << setw(20) << left << toStr(sq) << right << setw(12) << tc << endl; }
int main() { cout << "base inc id root sqr test count" << endl; auto st = chrono::system_clock::now(); for (base = 2; base <= maxBase; base++) doOne(); chrono::duration<double> et = chrono::system_clock::now() - st; cout << "\nComputation time was " << et.count() * 1000 << " milliseconds" << endl; return 0; }</lang>
- Output:
base inc id root sqr test count 2 1 10 100 0 3 1 22 2101 4 4 3 33 3201 2 5 1 2 243 132304 14 6 5 523 452013 20 7 6 1431 2450361 34 8 7 3344 13675420 41 9 4 11642 136802574 289 10 3 32043 1026753849 17 11 10 111453 1240A536789 1498 12 11 3966B9 124A7B538609 6883 13 1 3 3828943 10254773CA86B9 8242 14 13 3A9DB7C 10269B8C57D3A4 1330 15 14 1012B857 102597BACE836D4 4216 16 15 404A9D9B 1025648CFEA37BD9 18457 Computation time was 25.9016 milliseconds
C#
Based on the Visual Basic .NET version, plus it shortcuts some of the allIn() checks. When the numbers checked are below a threshold, not every digit needs to be checked, saving a little time. <lang csharp>using System; using System.Collections.Generic; using System.Numerics;
static class Program {
static byte Base, bmo, blim, ic; static DateTime st0; static BigInteger bllim, threshold; static HashSet<byte> hs = new HashSet<byte>(), o = new HashSet<byte>(); static string chars = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz|"; static List<BigInteger> limits; static string ms;
// convert BigInteger to string using current base
static string toStr(BigInteger b) {
string res = ""; BigInteger re; while (b > 0) {
b = BigInteger.DivRem(b, Base, out re); res = chars[(byte)re] + res;
} return res;
}
// check for a portion of digits, bailing if uneven
static bool allInQS(BigInteger b) {
BigInteger re; int c = ic; hs.Clear(); hs.UnionWith(o); while (b > bllim) {
b = BigInteger.DivRem(b, Base, out re);
hs.Add((byte)re); c += 1; if (c > hs.Count) return false;
} return true;
}
// check for a portion of digits, all the way to the end
static bool allInS(BigInteger b) {
BigInteger re; hs.Clear(); hs.UnionWith(o); while (b > bllim) {
b = BigInteger.DivRem(b, Base, out re); hs.Add((byte)re);
} return hs.Count == Base;
}
// check for all digits, bailing if uneven
static bool allInQ(BigInteger b) {
BigInteger re; int c = 0; hs.Clear(); while (b > 0) {
b = BigInteger.DivRem(b, Base, out re);
hs.Add((byte)re); c += 1; if (c > hs.Count) return false;
} return true;
}
// check for all digits, all the way to the end
static bool allIn(BigInteger b) {
BigInteger re; hs.Clear(); while (b > 0) {
b = BigInteger.DivRem(b, Base, out re); hs.Add((byte)re);
} return hs.Count == Base;
}
// parse a string into a BigInteger, using current base
static BigInteger to10(string s) {
BigInteger res = 0; foreach (char i in s) res = res * Base + chars.IndexOf(i);
return res;
}
// returns the minimum value string, optionally inserting extra digit
static string fixup(int n) {
string res = chars.Substring(0, Base); if (n > 0) res = res.Insert(n, n.ToString());
return "10" + res.Substring(2);
}
// checks the square against the threshold, advances various limits when needed
static void check(BigInteger sq) {
if (sq > threshold) {
o.Remove((byte)chars.IndexOf(ms[blim])); blim -= 1; ic -= 1;
threshold = limits[bmo - blim - 1]; bllim = to10(ms.Substring(0, blim + 1));
}
}
// performs all the caclulations for the current base
static void doOne() {
limits = new List<BigInteger>();
bmo = (byte)(Base - 1); byte dr = 0; if ((Base & 1) == 1) dr = (byte)(Base >> 1);
o.Clear(); blim = 0;
byte id = 0; int inc = 1; long i = 0; DateTime st = DateTime.Now; if (Base == 2) st0 = st;
byte[] sdr = new byte[bmo]; byte rc = 0; for (i = 0; i < bmo; i++) {
sdr[i] = (byte)((i * i) % bmo); rc += sdr[i] == dr ? (byte)1 : (byte)0;
sdr[i] += sdr[i] == 0 ? bmo : (byte)0;
} i = 0; if (dr > 0) {
id = Base;
for (i = 1; i <= dr; i++) if (sdr[i] >= dr) if (id > sdr[i]) id = sdr[i]; id -= dr;
i = 0;
} ms = fixup(id);
BigInteger sq = to10(ms); BigInteger rt = new BigInteger(Math.Sqrt((double)sq) + 1);
sq = rt * rt; if (Base > 9) {
for (int j = 1; j < Base; j++)
limits.Add(to10(ms.Substring(0, j) + new string(chars[bmo], Base - j + (rc > 0 ? 0 : 1))));
limits.Reverse(); while (sq < limits[0]) { rt++; sq = rt * rt; }
}
BigInteger dn = (rt << 1) + 1; BigInteger d = 1; if (Base > 3 && rc > 0) {
while (sq % bmo != dr) { rt += 1; sq += dn; dn += 2; } // alligns sq to dr
inc = bmo / rc;
if (inc > 1) { dn += rt * (inc - 2) - 1; d = inc * inc; }
dn += dn + d;
}
d <<= 1; if (Base > 9) {
blim = 0; while (sq < limits[bmo - blim - 1]) blim++; ic = (byte)(blim + 1);
threshold = limits[bmo - blim - 1];
if (blim > 0) for (byte j = 0; j <= blim; j++) o.Add((byte)chars.IndexOf(ms[j]));
if (blim > 0) bllim = to10(ms.Substring(0, blim + 1)); else bllim = 0;
if (Base > 5 && rc > 0)
do { if (allInQS(sq)) break; sq += dn; dn += d; i += 1; check(sq); } while (true);
else
do { if (allInS(sq)) break; sq += dn; dn += d; i += 1; check(sq); } while (true);
} else {
if (Base > 5 && rc > 0)
do { if (allInQ(sq)) break; sq += dn; dn += d; i += 1; } while (true);
else
do { if (allIn(sq)) break; sq += dn; dn += d; i += 1; } while (true);
} rt += i * inc;
Console.WriteLine("{0,3} {1,2} {2,2} {3,20} -> {4,-40} {5,10} {6,9:0.0000}s {7,9:0.0000}s",
Base, inc, (id > 0 ? chars.Substring(id, 1) : " "), toStr(rt), toStr(sq), i,
(DateTime.Now - st).TotalSeconds, (DateTime.Now - st0).TotalSeconds);
}
static void Main(string[] args) {
Console.WriteLine("base inc id root square" +
" test count time total");
for (Base = 2; Base <= 28; Base++) doOne();
Console.WriteLine("Elasped time was {0,8:0.00} minutes", (DateTime.Now - st0).TotalMinutes);
}
}</lang>
- Output:
base inc id root square test count time total 2 1 10 -> 100 0 0.0050s 0.0050s 3 1 22 -> 2101 4 0.0000s 0.0050s 4 3 33 -> 3201 2 0.0010s 0.0060s 5 1 2 243 -> 132304 14 0.0000s 0.0060s 6 5 523 -> 452013 20 0.0000s 0.0060s 7 6 1431 -> 2450361 34 0.0000s 0.0060s 8 7 3344 -> 13675420 41 0.0000s 0.0060s 9 4 11642 -> 136802574 289 0.0010s 0.0070s 10 3 32043 -> 1026753849 17 0.0050s 0.0120s 11 10 111453 -> 1240A536789 1498 0.0010s 0.0130s 12 11 3966B9 -> 124A7B538609 6883 0.0040s 0.0170s 13 1 3 3828943 -> 10254773CA86B9 8242 0.0439s 0.0609s 14 13 3A9DB7C -> 10269B8C57D3A4 1330 0.0010s 0.0619s 15 14 1012B857 -> 102597BACE836D4 4216 0.0020s 0.0638s 16 15 404A9D9B -> 1025648CFEA37BD9 18457 0.0100s 0.0738s 17 1 1 423F82GA9 -> 101246A89CGFB357ED 195112 0.2783s 0.3521s 18 17 44B482CAD -> 10236B5F8EG4AD9CH7 30440 0.0199s 0.3720s 19 6 1011B55E9A -> 10234DHBG7CI8F6A9E5 93021 0.0589s 0.4309s 20 19 49DGIH5D3G -> 1024E7CDI3HB695FJA8G 11310604 6.9833s 7.4142s 21 1 6 4C9HE5FE27F -> 1023457DG9HI8J6B6KCEAF 601843 1.0871s 8.5013s 22 21 4F94788GJ0F -> 102369FBGDEJ48CHI7LKA5 27804949 18.3290s 26.8302s 23 22 1011D3EL56MC -> 10234ACEDKG9HM8FBJIL756 17710217 11.4105s 38.2407s 24 23 4LJ0HDGF0HD3 -> 102345B87HFECKJNIGMDLA69 4266555 2.4763s 40.7171s 25 12 1011E145FHGHM -> 102345DOECKJ6GFB8LIAM7NH9 78092124 52.6831s 93.4012s 26 5 52K8N53BDM99K -> 1023458LO6IEMKG79FPCHNJDBA 402922568 287.9058s 381.3080s 27 26 1011F11E37OBJJ -> 1023458ELOMDHBIJFGKP7CQ9N6A 457555293 326.1714s 707.4794s 28 9 58A3CKP3N4CQD7 -> 1023456CGJBIRQEDHP98KMOAN7FL 749592976 508.4498s 1215.9292s Elasped time was 20.27 minutes
F#
The Task
<lang fsharp> // Nigel Galloway: May 21st., 2019 let fN g=let g=int64(sqrt(float(pown g (int(g-1L)))))+1L in (Seq.unfold(fun(n,g)->Some(n,(n+g,g+2L))))(g*g,g*2L+1L) let fG n g=Array.unfold(fun n->if n=0L then None else let n,g=System.Math.DivRem(n,g) in Some(g,n)) n let fL g=let n=set[0L..g-1L] in Seq.find(fun x->set(fG x g)=n) (fN g) let toS n g=let a=Array.concat [[|'0'..'9'|];[|'a'..'f'|]] in System.String(Array.rev(fG n g)|>Array.map(fun n->a.[(int n)])) [2L..16L]|>List.iter(fun n->let g=fL n in printfn "Base %d: %s² -> %s" n (toS (int64(sqrt(float g))) n) (toS g n)) </lang>
- Output:
Base 2: 10² -> 100 Base 3: 22² -> 2101 Base 4: 33² -> 3201 Base 5: 243² -> 132304 Base 6: 523² -> 452013 Base 7: 1431² -> 2450361 Base 8: 3344² -> 13675420 Base 9: 11642² -> 136802574 Base 10: 32043² -> 1026753849 Base 11: 111453² -> 1240a536789 Base 12: 3966b9² -> 124a7b538609 Base 13: 3828943² -> 10254773ca86b9 Base 14: 3a9db7c² -> 10269b8c57d3a4 Base 15: 1012b857² -> 102597bace836d4 Base 16: 404a9d9b² -> 1025648cfea37bd9
Using Factorial base numbers indexing permutations of a collection
On the discussion page for Factorial base numbers indexing permutations of a collection an anonymous contributor queries the value of Factorial base numbers indexing permutations of a collection. Well let's see him use an inverse Knuth shuffle to partially solve this task. This solution only applies to bases that do not require an extra digit. Still I think it's short and interesting.
Note that the minimal candidate is 1.0....0 as a factorial base number.
<lang fsharp>
// Nigel Galloway: May 30th., 2019
let fN n g=let g=n|>Array.rev|>Array.mapi(fun i n->(int64 n)*(pown g i))|>Array.sum
let n=int64(sqrt (float g)) in g=(n*n)
let fG g=lN([|yield 1; yield! Array.zeroCreate(g-2)|])|>Seq.map(fun n->lN2p n [|0..(g-1)|]) |> Seq.filter(fun n->fN n (int64 g)) printfn "%A" (fG 12|>Seq.head) // -> [|1; 2; 4; 10; 7; 11; 5; 3; 8; 6; 0; 9|] printfn "%A" (fG 14|>Seq.head) // -> [|1; 0; 2; 6; 9; 11; 8; 12; 5; 7; 13; 3; 10; 4|] </lang>
Go
This takes advantage of major optimizations described by Nigel Galloway and Thundergnat (inspired by initial pattern analysis by Hout) in the Discussion page and a minor optimization contributed by myself. <lang go>package main
import (
"fmt" "math/big" "strconv" "time"
)
const maxBase = 27 const minSq36 = "1023456789abcdefghijklmnopqrstuvwxyz" const minSq36x = "10123456789abcdefghijklmnopqrstuvwxyz"
var bigZero = new(big.Int) var bigOne = new(big.Int).SetUint64(1)
func containsAll(sq string, base int) bool {
var found [maxBase]byte
le := len(sq)
reps := 0
for _, r := range sq {
d := r - 48
if d > 38 {
d -= 39
}
found[d]++
if found[d] > 1 {
reps++
if le-reps < base {
return false
}
}
}
return true
}
func sumDigits(n, base *big.Int) *big.Int {
q := new(big.Int).Set(n)
r := new(big.Int)
sum := new(big.Int).Set(bigZero)
for q.Cmp(bigZero) == 1 {
q.QuoRem(q, base, r)
sum.Add(sum, r)
}
return sum
}
func digitalRoot(n *big.Int, base int) int {
root := new(big.Int)
b := big.NewInt(int64(base))
for i := new(big.Int).Set(n); i.Cmp(b) >= 0; i.Set(root) {
root.Set(sumDigits(i, b))
}
return int(root.Int64())
}
func minStart(base int) (string, uint64, int) {
nn := new(big.Int)
ms := minSq36[:base]
nn.SetString(ms, base)
bdr := digitalRoot(nn, base)
var drs []int
var ixs []uint64
for n := uint64(1); n < uint64(2*base); n++ {
nn.SetUint64(n * n)
dr := digitalRoot(nn, base)
if dr == 0 {
dr = int(n * n)
}
if dr == bdr {
ixs = append(ixs, n)
}
if n < uint64(base) && dr >= bdr {
drs = append(drs, dr)
}
}
inc := uint64(1)
if len(ixs) >= 2 && base != 3 {
inc = ixs[1] - ixs[0]
}
if len(drs) == 0 {
return ms, inc, bdr
}
min := drs[0]
for _, dr := range drs[1:] {
if dr < min {
min = dr
}
}
rd := min - bdr
if rd == 0 {
return ms, inc, bdr
}
if rd == 1 {
return minSq36x[:base+1], 1, bdr
}
ins := string(minSq36[rd])
return (minSq36[:rd] + ins + minSq36[rd:])[:base+1], inc, bdr
}
func main() {
start := time.Now()
var nb, nn big.Int
for n, k, base := uint64(2), uint64(1), 2; ; n += k {
if base > 2 && n%uint64(base) == 0 {
continue
}
nb.SetUint64(n)
sq := nb.Mul(&nb, &nb).Text(base)
if !containsAll(sq, base) {
continue
}
ns := strconv.FormatUint(n, base)
tt := time.Since(start).Seconds()
fmt.Printf("Base %2d:%15s² = %-27s in %8.3fs\n", base, ns, sq, tt)
if base == maxBase {
break
}
base++
ms, inc, bdr := minStart(base)
k = inc
nn.SetString(ms, base)
nb.Sqrt(&nn)
if nb.Uint64() < n+1 {
nb.SetUint64(n + 1)
}
if k != 1 {
for {
nn.Mul(&nb, &nb)
dr := digitalRoot(&nn, base)
if dr == bdr {
n = nb.Uint64() - k
break
}
nb.Add(&nb, bigOne)
}
} else {
n = nb.Uint64() - k
}
}
}</lang>
- Output:
Timings (in seconds) are for my Celeron @ 1.6GHz and should therefore be much faster on a more modern machine.
Base 2: 10² = 100 in 0.000s Base 3: 22² = 2101 in 0.000s Base 4: 33² = 3201 in 0.000s Base 5: 243² = 132304 in 0.000s Base 6: 523² = 452013 in 0.000s Base 7: 1431² = 2450361 in 0.000s Base 8: 3344² = 13675420 in 0.001s Base 9: 11642² = 136802574 in 0.001s Base 10: 32043² = 1026753849 in 0.001s Base 11: 111453² = 1240a536789 in 0.002s Base 12: 3966b9² = 124a7b538609 in 0.009s Base 13: 3828943² = 10254773ca86b9 in 0.018s Base 14: 3a9db7c² = 10269b8c57d3a4 in 0.020s Base 15: 1012b857² = 102597bace836d4 in 0.025s Base 16: 404a9d9b² = 1025648cfea37bd9 in 0.034s Base 17: 423f82ga9² = 101246a89cgfb357ed in 0.293s Base 18: 44b482cad² = 10236b5f8eg4ad9ch7 in 0.334s Base 19: 1011b55e9a² = 10234dhbg7ci8f6a9e5 in 0.459s Base 20: 49dgih5d3g² = 1024e7cdi3hb695fja8g in 16.017s Base 21: 4c9he5fe27f² = 1023457dg9hi8j6b6kceaf in 16.953s Base 22: 4f94788gj0f² = 102369fbgdej48chi7lka5 in 57.406s Base 23: 1011d3el56mc² = 10234acedkg9hm8fbjil756 in 83.505s Base 24: 4lj0hdgf0hd3² = 102345b87hfeckjnigmdla69 in 89.939s Base 25: 1011e145fhghm² = 102345doeckj6gfb8liam7nh9 in 211.535s Base 26: 52k8n53bdm99k² = 1023458lo6iemkg79fpchnjdba in 854.955s Base 27: 1011f11e37objj² = 1023458elomdhbijfgkp7cq9n6a in 1619.151s
It's possible to go beyond base 27 by using big.Int (rather than uint64) for N as well as N² though this takes about 16% longer to reach base 27 itself.
For example, to reach base 28 (the largest base shown in the OEIS table) we have: <lang go>package main
import (
"fmt" "math/big" "time"
)
const maxBase = 28
// etc
func main() {
start := time.Now()
var n, k, b, t, nn big.Int
n.SetUint64(2)
k.SetUint64(1)
b.SetUint64(2)
for base := 2; ; n.Add(&n, &k) {
if base > 2 && t.Rem(&n, &b).Cmp(bigZero) == 0 {
continue
}
sq := nn.Mul(&n, &n).Text(base)
if !containsAll(sq, base) {
continue
}
ns := n.Text(base)
tt := time.Since(start).Seconds()
fmt.Printf("Base %2d:%15s² = %-28s in %8.3fs\n", base, ns, sq, tt)
if base == maxBase {
break
}
base++
b.SetUint64(uint64(base))
ms, inc, bdr := minStart(base)
k.SetUint64(inc)
nn.SetString(ms, base)
n.Sqrt(&nn)
t.Add(&n, bigOne)
if n.Cmp(&t) == -1 {
n.Set(&t)
}
if inc != 1 {
for {
nn.Mul(&n, &n)
dr := digitalRoot(&nn, base)
if dr == bdr {
n.Sub(&n, &k)
break
}
n.Add(&n, bigOne)
}
} else {
n.Sub(&n, &k)
}
}
}</lang>
- Output:
Base 2: 10² = 100 in 0.000s Base 3: 22² = 2101 in 0.000s Base 4: 33² = 3201 in 0.001s Base 5: 243² = 132304 in 0.001s Base 6: 523² = 452013 in 0.001s Base 7: 1431² = 2450361 in 0.001s Base 8: 3344² = 13675420 in 0.001s Base 9: 11642² = 136802574 in 0.002s Base 10: 32043² = 1026753849 in 0.002s Base 11: 111453² = 1240a536789 in 0.004s Base 12: 3966b9² = 124a7b538609 in 0.016s Base 13: 3828943² = 10254773ca86b9 in 0.030s Base 14: 3a9db7c² = 10269b8c57d3a4 in 0.032s Base 15: 1012b857² = 102597bace836d4 in 0.038s Base 16: 404a9d9b² = 1025648cfea37bd9 in 0.052s Base 17: 423f82ga9² = 101246a89cgfb357ed in 0.369s Base 18: 44b482cad² = 10236b5f8eg4ad9ch7 in 0.421s Base 19: 1011b55e9a² = 10234dhbg7ci8f6a9e5 in 0.576s Base 20: 49dgih5d3g² = 1024e7cdi3hb695fja8g in 19.270s Base 21: 4c9he5fe27f² = 1023457dg9hi8j6b6kceaf in 20.375s Base 22: 4f94788gj0f² = 102369fbgdej48chi7lka5 in 68.070s Base 23: 1011d3el56mc² = 10234acedkg9hm8fbjil756 in 99.202s Base 24: 4lj0hdgf0hd3² = 102345b87hfeckjnigmdla69 in 106.909s Base 25: 1011e145fhghm² = 102345doeckj6gfb8liam7nh9 in 249.813s Base 26: 52k8n53bdm99k² = 1023458lo6iemkg79fpchnjdba in 999.026s Base 27: 1011f11e37objj² = 1023458elomdhbijfgkp7cq9n6a in 1880.265s Base 28: 58a3ckp3n4cqd7² = 1023456cgjbirqedhp98kmoan7fl in 3564.072s
JavaScript
<lang javascript>(() => {
'use strict';
// allDigitSquare :: Int -> Int
const allDigitSquare = base => {
const bools = replicate(base, false);
return untilSucc(
allDigitsUsedAtBase(base, bools),
ceil(sqrt(parseInt(
'10' + '0123456789abcdef'.slice(2, base),
base
)))
);
};
// allDigitsUsedAtBase :: Int -> [Bool] -> Int -> Bool
const allDigitsUsedAtBase = (base, bools) => n => {
// Fusion of representing the square of integer N at a given base
// with checking whether all digits of that base contribute to N^2.
// Sets the bool at a digit position to True when used.
// True if all digit positions have been used.
const ds = bools.slice(0);
let x = n * n;
while (x) {
ds[x % base] = true;
x = floor(x / base);
}
return ds.every(x => x)
};
// showBaseSquare :: Int -> String
const showBaseSquare = b => {
const q = allDigitSquare(b);
return justifyRight(2, ' ', str(b)) + ' -> ' +
justifyRight(8, ' ', showIntAtBase(b, digit, q, )) +
' -> ' + showIntAtBase(b, digit, q * q, );
};
// TEST -----------------------------------------------
const main = () => {
// 1-12 only - by 15 the squares are truncated by
// JS integer limits.
// Returning values through console.log –
// in separate events to avoid asynchronous disorder.
print('Smallest perfect squares using all digits in bases 2-12:\n')
print('Base Root Square')
print(showBaseSquare(2));
print(showBaseSquare(3));
print(showBaseSquare(4));
print(showBaseSquare(5));
print(showBaseSquare(6));
print(showBaseSquare(7));
print(showBaseSquare(8));
print(showBaseSquare(9));
print(showBaseSquare(10));
print(showBaseSquare(11));
print(showBaseSquare(12));
};
// GENERIC FUNCTIONS ----------------------------------
const
ceil = Math.ceil,
floor = Math.floor,
sqrt = Math.sqrt;
// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});
// digit :: Int -> Char
const digit = n =>
// Digit character for given integer.
'0123456789abcdef' [n];
// enumFromTo :: (Int, Int) -> [Int]
const enumFromTo = (m, n) =>
Array.from({
length: 1 + n - m
}, (_, i) => m + i);
// justifyRight :: Int -> Char -> String -> String
const justifyRight = (n, cFiller, s) =>
n > s.length ? (
s.padStart(n, cFiller)
) : s;
// print :: a -> IO () const print = x => console.log(x)
// quotRem :: Int -> Int -> (Int, Int)
const quotRem = (m, n) =>
Tuple(Math.floor(m / n), m % n);
// replicate :: Int -> a -> [a]
const replicate = (n, x) =>
Array.from({
length: n
}, () => x);
// showIntAtBase :: Int -> (Int -> Char) -> Int -> String -> String
const showIntAtBase = (base, toChr, n, rs) => {
const go = ([n, d], r) => {
const r_ = toChr(d) + r;
return 0 !== n ? (
go(Array.from(quotRem(n, base)), r_)
) : r_;
};
return 1 >= base ? (
'error: showIntAtBase applied to unsupported base'
) : 0 > n ? (
'error: showIntAtBase applied to negative number'
) : go(Array.from(quotRem(n, base)), rs);
};
// Abbreviation for quick testing - any 2nd arg interpreted as indent size
// sj :: a -> String
function sj() {
const args = Array.from(arguments);
return JSON.stringify.apply(
null,
1 < args.length && !isNaN(args[0]) ? [
args[1], null, args[0]
] : [args[0], null, 2]
);
}
// str :: a -> String const str = x => x.toString();
// untilSucc :: (Int -> Bool) -> Int -> Int
const untilSucc = (p, x) => {
// The first in a chain of successive integers
// for which p(x) returns true.
let v = x;
while (!p(v)) v = 1 + v;
return v;
};
// MAIN --- return main();
})();</lang>
- Output:
Smallest perfect squares using all digits in bases 2-12: Base Root Square 2 -> 10 -> 100 3 -> 22 -> 2101 4 -> 33 -> 3201 5 -> 243 -> 132304 6 -> 523 -> 452013 7 -> 1431 -> 2450361 8 -> 3344 -> 13675420 9 -> 11642 -> 136802574 10 -> 32043 -> 1026753849 11 -> 111453 -> 1240a536789 12 -> 3966b9 -> 124a7b538609
Julia
Runs in about 4 seconds with using occursin(). <lang julia>const num = "0123456789abcdef" hasallin(n, nums, b) = (s = string(n, base=b); all(x -> occursin(x, s), nums))
function squaresearch(base)
basenumerals = [c for c in num[1:base]]
highest = parse(Int, "10" * num[3:base], base=base)
for n in Int(trunc(sqrt(highest))):highest
if hasallin(n * n, basenumerals, base)
return n
end
end
end
println("Base Root N") for b in 2:16
n = squaresearch(b) println(lpad(b, 3), lpad(string(n, base=b), 10), " ", string(n * n, base=b))
end
</lang>
- Output:
Base Root N 2 10 100 3 22 2101 4 33 3201 5 243 132304 6 523 452013 7 1431 2450361 8 3344 13675420 9 11642 136802574 10 32043 1026753849 11 111453 1240a536789 12 3966b9 124a7b538609 13 3828943 10254773ca86b9 14 3a9db7c 10269b8c57d3a4 15 1012b857 102597bace836d4 16 404a9d9b 1025648cfea37bd9
Pascal
Using an array of digits to base n, to get rid of base conversions.
Starting value equals squareroot of smallest value containing all digits to base.
Than brute force.
Try it online!
<lang pascal>program project1;
//Find the smallest number n to base b, so that n*n includes all
//digits of base b
{$IFDEF FPC}{$MODE DELPHI}{$ENDIF}
uses
sysutils;
const
charSet : array[0..36] of char ='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ';
type
tNumtoBase = record
ntb_dgt : array[0..31-4] of byte;
ntb_cnt,
ntb_bas : Word;
end;
var
Num, sqr2B, deltaNum : tNumtoBase;
function Minimal_n(base:NativeUint):Uint64; //' 1023456789ABCDEFGHIJ...' var
i : NativeUint;
Begin
result := base; // aka '10'
IF base > 2 then
For i := 2 to base-1 do
result := result*base+i;
result := trunc(sqrt(result)+0.99999);
end;
procedure Conv2num(var num:tNumtoBase;n:Uint64;base:NativeUint); var
quot :UInt64; i :NativeUint;
Begin
i := 0;
repeat
quot := n div base;
Num.ntb_dgt[i] := n-quot*base;
n := quot;
inc(i);
until n = 0;
Num.ntb_cnt := i;
Num.ntb_bas := base;
//clear upper digits
For i := i to high(tNumtoBase.ntb_dgt) do
Num.ntb_dgt[i] := 0;
end;
procedure OutNum(const num:tNumtoBase); var
i : NativeInt;
Begin
with num do
Begin
For i := 17-ntb_cnt-1 downto 0 do
write(' ');
For i := ntb_cnt-1 downto 0 do
write(charSet[ntb_dgt[i]]);
end;
end;
procedure IncNumBig(var add1:tNumtoBase;n:NativeUInt); //prerequisites //bases are the same,delta : NativeUint var
i,s,b,carry : NativeInt;
Begin
b := add1.ntb_bas; i := 0; carry := 0; while n > 0 do Begin s := add1.ntb_dgt[i]+carry+ n MOD b; carry := Ord(s>=b); s := s- (-carry AND b); add1.ntb_dgt[i] := s; n := n div b; inc(i); end; while carry <> 0 do Begin s := add1.ntb_dgt[i]+carry; carry := Ord(s>=b); s := s- (-carry AND b); add1.ntb_dgt[i] := s; inc(i); end;
IF add1.ntb_cnt < i then add1.ntb_cnt := i;
end;
procedure IncNum(var add1:tNumtoBase;carry:NativeInt); //prerequisites: bases are the same, carry==delta < base var
i,s,b : NativeInt;
Begin
b := add1.ntb_bas; i := 0; while carry <> 0 do Begin s := add1.ntb_dgt[i]+carry; carry := Ord(s>=b); s := s- (-carry AND b); add1.ntb_dgt[i] := s; inc(i); end; IF add1.ntb_cnt < i then add1.ntb_cnt := i;
end;
procedure AddNum(var add1,add2:tNumtoBase); //prerequisites //bases are the same,add1>add2, add1 <= add1+add2; var
i,carry,s,b : NativeInt;
Begin
b := add1.ntb_bas; carry := 0; For i := 0 to add2.ntb_cnt-1 do begin s := add1.ntb_dgt[i]+add2.ntb_dgt[i]+carry; carry := Ord(s>=b); s := s- (-carry AND b); add1.ntb_dgt[i] := s; end; i := add2.ntb_cnt; while carry = 1 do Begin s := add1.ntb_dgt[i]+carry; carry := Ord(s>=b); // remove of if s>b then by bit-twiddling s := s- (-carry AND b); add1.ntb_dgt[i] := s; inc(i); end; IF add1.ntb_cnt < i then add1.ntb_cnt := i;
end;
procedure Test(base:NativeInt); var
n : Uint64; i,j,TestSet : NativeInt;
Begin
write(base:5);
n := Minimal_n(base);
Conv2num(sqr2B,n*n,base);
Conv2num(Num,n,base);
deltaNum := num;
AddNum(deltaNum,deltaNum);
IncNum(deltaNum,1);
i := 0;
repeat
//count used digits
TestSet := 0;
For j := sqr2B.ntb_cnt-1 downto 0 do
TestSet := TestSet OR (1 shl sqr2B.ntb_dgt[j]);
inc(TestSet);
IF (1 shl base)=TestSet then
BREAK;
//next square number
AddNum(sqr2B,deltaNum);
IncNum(deltaNum,2);
inc(i);
until false;
IncNumBig(num,i);
OutNum(Num);
OutNum(sqr2B);
Writeln(i:14);
end;
var
T0: TDateTime; base :nativeInt;
begin
T0 := now;
writeln('base n square(n) Testcnt');
For base := 2 to 16 do
Test(base);
writeln((now-T0)*86400:10:3);
{$IFDEF WINDOWS}readln;{$ENDIF}
end.</lang>
- Output:
base n square(n) Testcnt
2 10 100 0
3 22 2101 4
4 33 3201 6
5 243 132304 46
6 523 452013 103
7 1431 2450361 209
8 3344 13675420 288
9 11642 136802574 1156
10 32043 1026753849 51
11 111453 1240A536789 14983
12 3966B9 124A7B538609 75713
13 3828943 10254773CA86B9 12668112
14 3A9DB7C 10269B8C57D3A4 17291
15 1012B857 102597BACE836D4 59026
16 404A9D9B 1025648CFEA37BD9 276865
0.401
Inserted nearly all optimizations found by Hout and Nigel Galloway
I use now gmp to calculate the start values.Check Chai Wah Wu list on oeis.org/A260182
Try it online!
The runtime is on my PC AMD Ryzen 3 2200G Win 10 1903 .
<lang pascal>program project1;
//Find the smallest number n to base b, so that n*n includes all
//digits of base b aka pandigital
{$IFDEF FPC}
//{$R+,O+}
{$MODE DELPHI}
{$Optimization ON}
{$CODEALIGN proc=1,loop=1}
{$ENDIF} uses
SysUtils, gmp;// to calculate start values
const {$ALIGN 32}
cOr_Mask : array[0..63] of Uint64 =
(1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,
32768,65536,131072,262144,524288,1048576,2097152,4194304,
8388608,16777216,33554432,67108864,134217728,268435456,
536870912,1073741824,2147483648,4294967296,8589934592,
17179869184,34359738368,68719476736,137438953472,
274877906944,549755813888,1099511627776,2199023255552,
4398046511104,8796093022208,17592186044416,35184372088832,
70368744177664,140737488355328,281474976710656,
562949953421312,1125899906842624,2251799813685248,
4503599627370496,9007199254740992,18014398509481984,
36028797018963968,72057594037927936,144115188075855872,
288230376151711744,576460752303423488,1152921504606846976,
2305843009213693952,4611686018427387904,9223372036854775808);
charSet: array[0..62] of char = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz';
type
tConvert = record
newval :word;
newcry :word;
end;
tArrConvert = array[0..62*62] of tConvert;
tpArrConv = ^tArrConvert;
tNumtoBase = packed record ntb_dgt: array[0..63 - 8] of byte; ntb_cnt, ntb_bas: Uint32; end;
tDgtRootSqr = packed record drs_List: array[0..61] of byte; drs_bas: byte; drs_Sol: byte; drs_SolCnt: byte; drs_Dgt2Add:byte; drs_NeedsOneMoreDigit: boolean; end;
var {$ALIGN 32}
Convert: tArrConvert;
{$ALIGN 32}
Num, sqr2B, deltaNextSqr, delta: tNumtoBase;
{$ALIGN 32}
DgtRtSqr: tDgtRootSqr;
{$ALIGN 8}
T0, T1: TDateTime;
procedure InitConvert(base:NativeInt);
var
i : integer;
Begin
For i := 0 to Base-1 do
Begin
with convert[i] do
Begin
newval := i;
newcry :=0;
end;
with convert[i+base] do
Begin
newval := i;
newcry := 1;
end;
end;
end;
procedure OutNum(const num: tNumtoBase);
var
i: NativeInt;
begin
with num do
begin
for i := ntb_cnt - 1 downto 0 do
Write(charSet[ntb_dgt[i]]);
end;
Write(' ');
end;
procedure OutNumSqr;
Begin
write(' Num ');
OutNum(Num);
write(' sqr ');
OutNum(sqr2B);
writeln;
end;
procedure OutIndex(i: NativeUint);
var
s: string[127];
p: NativeInt;
begin
Write(#13, i div 1000000: 10, ' Mio ');
//check last digit sqr(num) mod base must be last digit of sqrnumber
if (sqr(Num.ntb_dgt[0]) mod Num.ntb_bas) <> sqr2B.ntb_dgt[0] then
begin
with Num do
begin
p := 1;
setlength(s, ntb_cnt);
for i := ntb_cnt - 1 downto 0 do
begin
s[p] := charSet[ntb_dgt[i]];
Inc(p);
end;
end;
s[p] := ' ';
Inc(p);
with sqr2B do
begin
setlength(s, length(s) + ntb_cnt);
for i := ntb_cnt - 1 downto 0 do
begin
s[p] := charSet[ntb_dgt[i]];
Inc(p);
end;
end;
writeln(s);
end;
end;
function getDgtRtNum(const num: tNumtoBase): NativeInt;
var
i: NativeInt;
begin
with num do
begin
Result := 0;
for i := 0 to num.ntb_cnt - 1 do
Inc(Result, ntb_dgt[i]);
Result := Result mod (ntb_bas - 1);
end;
end;
procedure CalcDgtRootSqr(base: NativeUInt);
var
i,j,MinAdd: NativeUInt;
begin
with DgtRtSqr do
begin
//pandigtal digital root (sum all digits of base) mod (base-1)
drs_bas := base;
if Odd(base) then
drs_Sol := base div 2
else
drs_Sol := 0;
base := base - 1;
//calc which dgt root the square of the number will become
for i := 0 to base - 1 do
drs_List[i] := (i * i) mod base;
//searching for solution
drs_SolCnt := 0;
for i := 0 to base - 1 do
if drs_List[i] = drs_Sol then
Inc(drs_SolCnt);
//if not found then NeedsOneMoreDigit
drs_NeedsOneMoreDigit := drs_SolCnt = 0;
IF drs_NeedsOneMoreDigit then
Begin
MinAdd := Base;
for j := 1 to drs_Sol do
for i := 1 to MinAdd do
IF drs_List[j] = (drs_Sol+i) MOD BASE then
if i < MinAdd then
MinAdd := i;
drs_Dgt2Add := MinAdd;
writeln('insert ',MinAdd);
end;
end;
end;
procedure conv_ui_num(base: NativeUint; ui: Uint64; var Num: tNumtoBase);
var
i: NativeUInt;
begin
for i := 0 to high(tNumtoBase.ntb_dgt) do
Num.ntb_dgt[i] := 0;
with num do
begin
ntb_bas := base;
ntb_cnt := 0;
if ui = 0 then
EXIT;
i := 0;
repeat
ntb_dgt[i] := ui mod base;
ui := ui div base;
Inc(i);
until ui = 0;
ntb_cnt := i;
end;
end;
procedure conv2Num(base: NativeUint; var Num: tNumtoBase; var s: mpz_t);
var
i: NativeUInt;
begin
for i := 0 to high(tNumtoBase.ntb_dgt) do
Num.ntb_dgt[i] := 0;
with num do
begin
ntb_bas := base;
i := 0;
repeat
ntb_dgt[i] := mpz_tdiv_q_ui(s, s, base);
Inc(i);
until mpz_cmp_ui(s, 0) = 0;
ntb_cnt := i;
end;
end;
procedure StartValueCreate(base: NativeUInt); //create the lowest pandigital number "102345...Base-1 " //calc sqrt +1 and convert n new format. var sv_sqr, sv: mpz_t; k,dblDgt: NativeUint;
begin mpz_init(sv); mpz_init(sv_sqr);
mpz_init_set_si(sv_sqr, base);//"10" CalcDgtRootSqr(base);
if DgtRtSqr.drs_NeedsOneMoreDigit then
begin
dblDgt := DgtRtSqr.drs_Dgt2Add;
IF dblDgt= 1 then
Begin
For k := 1 to base-1 do
Begin
mpz_mul_ui(sv_sqr, sv_sqr, base);
mpz_add_ui(sv_sqr, sv_sqr, k);
end;
end
else
Begin
For k := 2 to dblDgt do
Begin
mpz_mul_ui(sv_sqr, sv_sqr, base);
mpz_add_ui(sv_sqr, sv_sqr, k);
end;
For k := dblDgt to base-1 do
Begin
mpz_mul_ui(sv_sqr, sv_sqr, base);
mpz_add_ui(sv_sqr, sv_sqr, k);
end;
end
end
else
begin
For k := 2 to base-1 do
begin
mpz_mul_ui(sv_sqr, sv_sqr, base);
mpz_add_ui(sv_sqr, sv_sqr, k);
end;
end;
mpz_sqrt(sv, sv_sqr); mpz_add_ui(sv, sv, 1); mpz_mul(sv_sqr, sv, sv);
if base < 63 then
begin
conv2Num(base, Num, sv);
conv2Num(base, sqr2B, sv_sqr);
end;
mpz_clear(sv_sqr);
mpz_clear(sv);
end;
procedure IncNumBig(var add1: tNumtoBase; n: Uint64);
var
i, s, b, carry: NativeUInt;
begin
b := add1.ntb_bas;
i := 0;
carry := 0;
while n > 0 do
begin
s := add1.ntb_dgt[i] + carry + n mod b;
carry := Ord(s >= b);
s := s - (-carry and b);
add1.ntb_dgt[i] := s;
n := n div b;
Inc(i);
end;
while carry <> 0 do
begin
s := add1.ntb_dgt[i] + carry;
carry := Ord(s >= b);
s := s - (-carry and b);
add1.ntb_dgt[i] := s;
Inc(i);
end;
if add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
procedure IncNum(var add1: tNumtoBase; carry: NativeUInt);inline;
//prerequisites carry < base
var
i, s, b: NativeUInt;
begin
b := add1.ntb_bas;
i := 0;
while carry <> 0 do
begin
s := add1.ntb_dgt[i] + carry;
carry := Ord(s >= b);
s := s - (-carry and b);
add1.ntb_dgt[i] := s;
Inc(i);
end;
if add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
procedure AddNum(var add1, add2: tNumtoBase); //add1 <= add1+add2; //prerequisites bases are the same,add1>=add2( cnt ), var
// pC : tpArrConv;
i: NativeInt; base,s,carry: NativeUInt; begin base := add1.ntb_bas;
// pC := @Convert;
carry := 0;
for i := 0 to add2.ntb_cnt - 1 do
begin
s := add1.ntb_dgt[i] + add2.ntb_dgt[i]+carry;
carry := Ord(s >= base);
s := s - (-carry and base);
add1.ntb_dgt[i] := s;
{
with pC[add1.ntb_dgt[i] + add2.ntb_dgt[i]+carry] do
Begin
add1.ntb_dgt[i] := newval;
carry := newcry;
end;
* }
end;
i := add2.ntb_cnt;
while carry = 1 do
begin
s := add1.ntb_dgt[i] +carry;
carry := Ord(s >= base);
s := s - (-carry and base);
add1.ntb_dgt[i] := s;
{
with pC[add1.ntb_dgt[i]+carry] do
Begin
add1.ntb_dgt[i] := newval;
carry := newcry;
end;}
Inc(i);
end;
if add1.ntb_cnt < i then
add1.ntb_cnt := i;
end;
function TestRun(base:NativeInt):NativeInt;
var
pMask : pUint64;
pSqrNum,pdeltaNextSqr : ^tNumtoBase;
TestSet,TestSetComplete: Uint64;
j: NativeInt;
begin
TestSetComplete := Uint64(1) shl base - 1;
result := 0;
pSqrNum := @sqr2B;
pdeltaNextSqr := @deltaNextSqr;
pMask := @cOr_Mask;
repeat
TestSet := 0;
//next square number
AddNum(pSqrNum^, pdeltaNextSqr^);
AddNum(pdeltaNextSqr^, delta);
//check used digits
for j := 0 to pSqrNum^.ntb_cnt - 1 do
TestSet := TestSet or pMask[pSqrNum^.ntb_dgt[j]];
Inc(result);
until TestSetComplete = TestSet;
end;
procedure Test(base:NativeInt);
var
deltaCnt, TestSet,MyOne, TestSetComplete: Uint64;
i, j: NativeInt;
begin
//InitConvert(base);
T0 := now;
StartValueCreate(base);
deltaNextSqr := num;
AddNum(deltaNextSqr, deltaNextSqr);
IncNum(deltaNextSqr, 1);
deltaCnt := 1;
if (Base > 3) and not (DgtRtSqr.drs_NeedsOneMoreDigit) then
begin
//Find first number which can get the solution
with dgtrtsqr do
while drs_List[getDgtRtNum(num)] <> drs_sol do
begin
IncNum(num, 1);
AddNum(sqr2B, deltaNextSqr);
IncNum(deltaNextSqr, 2);
end;
deltaCnt := (Base - 1) div DgtRtSqr.drs_SolCnt;
//j*num
deltaNextSqr := num;
for i := 2 to deltaCnt do
AddNum(deltaNextSqr, num);
AddNum(deltaNextSqr, deltaNextSqr);
IncNumBig(deltaNextSqr, deltaCnt * deltaCnt);
end;
conv_ui_num(base, 2 * deltaCnt * deltaCnt, delta);
writeln('Base ', base, ' test every ', deltaCnt);
Write('Start :');OutNumSqr;
i := 0;
MyOne := 1;
TestSetComplete := MyOne shl base - 1;
//count used digits
TestSet := 0;
for j := sqr2B.ntb_cnt - 1 downto 0 do
TestSet := TestSet or (MyOne shl sqr2B.ntb_dgt[j]);
if TestSetComplete <> TestSet then
Begin
if deltaCnt = 1 then
begin
repeat
//next square number
AddNum(sqr2B, deltaNextSqr);
IncNum(deltaNextSqr, 2);
Inc(i);
TestSet := 0;
for j := sqr2B.ntb_cnt - 1 downto 0 do
TestSet := TestSet or (MyOne shl sqr2B.ntb_dgt[j]);
if TestSetComplete = TestSet then BREAK;
// IncNum(num, 1);if i and (1 shl 23 - 1) = 0 then OutIndex(i);
until false;
end
else
begin
i := TestRun(base);
end;
IncNumBig(num,i*deltaCnt);
end;
T1 := now;
Write('Result :');OutNumSqr;
Writeln(#13,(T1 - t0) * 86400: 9: 3, ' s Testcount : ', i);
end;
var
T: TDateTime; base : NativeUint;
begin
writeln('base n square(n) Testcnt');
T:= now;
For base := 2 to 32 do
Test(base);
writeln('completed in ',(now - T) * 86400: 0: 3, ' seconds');
{$IFDEF WINDOWS}
readln;
{$ENDIF}
end.</lang>
- Output:
base n square(n) Testcnt
Base 2 test every 1
Start : Num 10 sqr 100
Result : Num 10 sqr 100
0.000 s Testcount : 0
Base 3 test every 1
Start : Num 11 sqr 121
Result : Num 22 sqr 2101
0.000 s Testcount : 4
Base 4 test every 3
Start : Num 21 sqr 1101
Result : Num 33 sqr 3201
0.000 s Testcount : 2
insert 2
Base 5 test every 1
Start : Num 214 sqr 102411
Result : Num 243 sqr 132304
0.000 s Testcount : 14
Base 6 test every 5
Start : Num 235 sqr 105441
Result : Num 523 sqr 452013
0.000 s Testcount : 20
Base 7 test every 6
Start : Num 1020 sqr 1040400
Result : Num 1431 sqr 2450361
0.000 s Testcount : 34
Base 8 test every 7
Start : Num 2705 sqr 10244631
Result : Num 3344 sqr 13675420
0.000 s Testcount : 41
Base 9 test every 4
Start : Num 10117 sqr 102363814
Result : Num 11642 sqr 136802574
0.000 s Testcount : 289
Base 10 test every 3
Start : Num 31992 sqr 1023488064
Result : Num 32043 sqr 1026753849
0.000 s Testcount : 17
Base 11 test every 10
Start : Num 101175 sqr 10235267A63
Result : Num 111453 sqr 1240A536789
0.000 s Testcount : 1498
Base 12 test every 11
Start : Num 35A924 sqr 102345A32554
Result : Num 3966B9 sqr 124A7B538609
0.000 s Testcount : 6883
insert 3
Base 13 test every 1
Start : Num 3824C73 sqr 10233460766739
Result : Num 3828943 sqr 10254773CA86B9
0.015 s Testcount : 8242
Base 14 test every 13
Start : Num 3A9774C sqr 1023457801D984
Result : Num 3A9DB7C sqr 10269B8C57D3A4
0.000 s Testcount : 1330
Base 15 test every 14
Start : Num 10119108 sqr 1023456BA5BA144
Result : Num 1012B857 sqr 102597BACE836D4
0.000 s Testcount : 4216
Base 16 test every 15
Start : Num 40466424 sqr 1023456CEADC2510
Result : Num 404A9D9B sqr 1025648CFEA37BD9
0.000 s Testcount : 18457
insert 1
Base 17 test every 1
Start : Num 423F5E486 sqr 101234567967G80FD2
Result : Num 423F82GA9 sqr 101246A89CGFB357ED
0.016 s Testcount : 195112
Base 18 test every 17
Start : Num 44B433H7F sqr 102345679E6908HD69
Result : Num 44B482CAD sqr 10236B5F8EG4AD9CH7
0.000 s Testcount : 30440
Base 19 test every 6
Start : Num 1011B10789 sqr 102345678I39A8G87F5
Result : Num 1011B55E9A sqr 10234DHBG7CI8F6A9E5
0.000 s Testcount : 93021
Base 20 test every 19
Start : Num 49DDBE2JA0 sqr 102345678D5CCEH05000
Result : Num 49DGIH5D3G sqr 1024E7CDI3HB695FJA8G
0.390 s Testcount : 11310604
insert 6
Base 21 test every 1
Start : Num 4C9HE5CC2DB sqr 10234566789GK362F7BGIG
Result : Num 4C9HE5FE27F sqr 1023457DG9HI8J6B6KCEAF
0.032 s Testcount : 601843
Base 22 test every 21
Start : Num 4F942523JL0 sqr 1023456789HL35DJ1I4100
Result : Num 4F94788GJ0F sqr 102369FBGDEJ48CHI7LKA5
0.984 s Testcount : 27804949
Base 23 test every 22
Start : Num 1011D108L54M sqr 1023456789C7F59L30C8ED1
Result : Num 1011D3EL56MC sqr 10234ACEDKG9HM8FBJIL756
0.594 s Testcount : 17710217
Base 24 test every 23
Start : Num 4LJ0HD4763F6 sqr 1023456789AC9NJIL6HG54DC
Result : Num 4LJ0HDGF0HD3 sqr 102345B87HFECKJNIGMDLA69
0.172 s Testcount : 4266555
Base 25 test every 12
Start : Num 1011E109GHMMM sqr 1023456789ABD5AHDHG370GC9
Result : Num 1011E145FHGHM sqr 102345DOECKJ6GFB8LIAM7NH9
2.969 s Testcount : 78092125
Base 26 test every 5
Start : Num 52K8N4MNP7AME sqr 1023456789ABEBLL1L0F3FG7PE
Result : Num 52K8N53BDM99K sqr 1023458LO6IEMKG79FPCHNJDBA
14.251 s Testcount : 402922568
Base 27 test every 26
Start : Num 1011F10AB5HL71 sqr 1023456789ABD6808CDF1LQ7AE1
Result : Num 1011F11E37OBJJ sqr 1023458ELOMDHBIJFGKP7CQ9N6A
17.705 s Testcount : 457555293
Base 28 test every 9
Start : Num 58A3CKOHN4IK4L sqr 1023456789ABCGJDO8M4JG8HMMFL
Result : Num 58A3CKP3N4CQD7 sqr 1023456CGJBIRQEDHP98KMOAN7FL
32.530 s Testcount : 749593054
insert 2
Base 29 test every 1
Start : Num 5BAEFC5QHESPCLA sqr 10223456789ABCDKM4JI4S470KCSHD
Result : Num 5BAEFC62RGS0KJF sqr 102234586REOSIGJD9PCF7HBLKANQM
4921.784 s Testcount : 92238034003
Base 30 test every 29
Start : Num 5EF7R2P77FFPBN5 sqr 1023456789ABCDHNHROTMC0MS6RGKP
Result : Num 5EF7R2POS9MQRN7 sqr 1023456DMAPECBQOLSITK9FR87GHNJ
571.767 s Testcount : 13343410738
Base 31 test every 30
Start : Num 1011H10BS64GFL76 sqr 1023456789ABCDF03FNNQ29H0ULION5
Result : Num 1011H10CDMAUP44O sqr 10234568ABQUJGCNFP7KEM9RHDLTSOI
651.246 s Testcount : 15152895679
Base 32 test every 31
Start : Num 5L6HID7BTGM6RUAA sqr 1023456789ABCDEMULAP8DRPBULSA2B4
Result : Num 5L6HID7BVGE2CIEC sqr 102345678VS9CMJDRAIOPLHNFQETBUKG
111.467 s Testcount : 2207946558
..
Base 33 test every 8
1011I10CLWWNS6SKS 102345678THKFAERNWJGDOSQ9BCIUVMLP
5210.788 s Testcount : 53808573863
Base 34 test every 33
5SEMXRII42NG8AKSL 102345679JIESRPA8BLCVKDNMHUFTGOQWX
28900.032 s Testcount : 205094427126
base 35 test every 34
1011J10DEFW1QTVBXR 102345678RUEPV9KGQIWFOBAXCNSLDMYJHT
48673.607 s Testcount : 614575698110
base 36 stopped after 3.3e12 tests ..
base 37 not tested
Base 38 test every 37
66FVHSMH0P60WK173YQ 1023456789DRTAINWaFJCHLYMQPGEBZVOKXSbU
114611.561 s Testcount : 1,242,398,966,051
Perl
<lang perl>use strict; use warnings; use feature 'say'; use ntheory qw/fromdigits todigitstring/; use utf8; binmode('STDOUT', 'utf8');
sub first_square {
my $n = shift;
my $sr = substr('1023456789abcdef',0,$n);
my $r = int fromdigits($sr, $n) ** .5;
my @digits = reverse split , $sr;
TRY: while (1) {
my $sq = $r * $r;
my $cnt = 0;
my $s = todigitstring($sq, $n);
my $i = scalar @digits;
for (@digits) {
$r++ and redo TRY if (-1 == index($s, $_)) || ($i-- + $cnt < $n);
last if $cnt++ == $n;
}
return sprintf "Base %2d: %10s² == %s", $n, todigitstring($r, $n),
todigitstring($sq, $n);
}
}
say "First perfect square with N unique digits in base N: "; say first_square($_) for 2..16;</lang>
- Output:
First perfect square with N unique digits in base N: Base 2: 10² == 100 Base 3: 22² == 2101 Base 4: 33² == 3201 Base 5: 243² == 132304 Base 6: 523² == 452013 Base 7: 1431² == 2450361 Base 8: 3344² == 13675420 Base 9: 11642² == 136802574 Base 10: 32043² == 1026753849 Base 11: 111453² == 1240a536789 Base 12: 3966b9² == 124a7b538609 Base 13: 3828943² == 10254773ca86b9 Base 14: 3a9db7c² == 10269b8c57d3a4 Base 15: 1012b857² == 102597bace836d4 Base 16: 404a9d9b² == 1025648cfea37bd9
Alternative solution:
<lang perl>use strict; use warnings; use ntheory qw(:all); use List::Util qw(uniq);
sub first_square {
my ($base) = @_;
my $start = sqrtint(fromdigits([1, 0, 2 .. $base-1], $base));
for (my $k = $start ; ; ++$k) {
if (uniq(todigits($k * $k, $base)) == $base) {
return $k * $k;
}
}
}
foreach my $n (2 .. 16) {
my $s = first_square($n);
printf("Base %2d: %10s² == %s\n", $n,
todigitstring(sqrtint($s), $n), todigitstring($s, $n));
}</lang>
Perl 6
As long as you have the patience, this will work for bases 2 through 36.
Bases 2 through 19 finish quickly, (about 10 seconds on my system), 20 takes a while, 21 is pretty fast, 22 is glacial. 23 through 26 takes several hours.
Use analytical start value filtering based on observations by Hout++ and Nigel Galloway++ on the discussion page.
<lang perl6>#`[
Only search square numbers that have at least N digits; smaller could not possibly match.
Only bother to use analytics for large N. Finesse takes longer than brute force for small N.
]
unit sub MAIN ($timer = False);
sub first-square (Int $n) {
my @start = flat '1', '0', (2 ..^ $n)».base: $n;
if $n > 10 { # analytics
my $root = digital-root( @start.join, :base($n) );
my @roots = (2..$n).map(*²).map: { digital-root($_.base($n), :base($n) ) };
if $root ∉ @roots {
my $offset = min(@roots.grep: * > $root ) - $root;
@start[1+$offset] = $offset ~ @start[1+$offset];
}
}
my $start = @start.join.parse-base($n).sqrt.ceiling;
my @digits = reverse (^$n)».base: $n;
my $sq;
my $now = now;
my $time = 0;
my $sr;
for $start .. * {
$sq = .²;
my $s = $sq.base($n);
my $f;
$f = 1 and last unless $s.contains: $_ for @digits;
if $timer && $n > 19 && $_ %% 1_000_000 {
$time += now - $now;
say "N $n: {$_}² = $sq <$s> : {(now - $now).round(.001)}s" ~
" : {$time.round(.001)} elapsed";
$now = now;
}
next if $f;
$sr = $_;
last
}
sprintf( "Base %2d: %13s² == %-30s", $n, $sr.base($n), $sq.base($n) ) ~
($timer ?? ($time + now - $now).round(.001) !! );
}
sub digital-root ($root is copy, :$base = 10) {
$root = $root.comb.map({:36($_)}).sum.base($base) while $root.chars > 1;
$root.parse-base($base);
}
say "First perfect square with N unique digits in base N: "; say .&first-square for flat
2 .. 12, # required 13 .. 16, # optional 17 .. 19, # stretch 20, # slow 21, # pretty fast 22, # very slow 23, # don't hold your breath 24, # slow but not too terrible 25, # very slow 26, # "
- </lang>
- Output:
First perfect square with N unique digits in base N: Base 2: 10² == 100 Base 3: 22² == 2101 Base 4: 33² == 3201 Base 5: 243² == 132304 Base 6: 523² == 452013 Base 7: 1431² == 2450361 Base 8: 3344² == 13675420 Base 9: 11642² == 136802574 Base 10: 32043² == 1026753849 Base 11: 111453² == 1240A536789 Base 12: 3966B9² == 124A7B538609 Base 13: 3828943² == 10254773CA86B9 Base 14: 3A9DB7C² == 10269B8C57D3A4 Base 15: 1012B857² == 102597BACE836D4 Base 16: 404A9D9B² == 1025648CFEA37BD9 Base 17: 423F82GA9² == 101246A89CGFB357ED Base 18: 44B482CAD² == 10236B5F8EG4AD9CH7 Base 19: 1011B55E9A² == 10234DHBG7CI8F6A9E5 Base 20: 49DGIH5D3G² == 1024E7CDI3HB695FJA8G Base 21: 4C9HE5FE27F² == 1023457DG9HI8J6B6KCEAF Base 22: 4F94788GJ0F² == 102369FBGDEJ48CHI7LKA5 Base 23: 1011D3EL56MC² == 10234ACEDKG9HM8FBJIL756 Base 24: 4LJ0HDGF0HD3² == 102345B87HFECKJNIGMDLA69 Base 25: 1011E145FHGHM² == 102345DOECKJ6GFB8LIAM7NH9 Base 26: 52K8N53BDM99K² == 1023458LO6IEMKG79FPCHNJDBA
Python
<lang python>Perfect squares using every digit in a given base.
from itertools import (count, dropwhile, repeat) from math import (ceil, sqrt) from time import time
- allDigitSquare :: Int -> Int -> Int
def allDigitSquare(base, above):
The lowest perfect square which
requires all digits in the given base.
bools = list(repeat(True, base))
return next(dropwhile(missingDigitsAtBase(base, bools), count(
max(above, ceil(sqrt(int('10' + '0123456789abcdef'[2:base], base))))
)))
- missingDigitsAtBase :: Int -> [Bool] -> Int -> Bool
def missingDigitsAtBase(base, bools):
Fusion of representing the square of integer N at a given base
with checking whether all digits of that base contribute to N^2.
Clears the bool at a digit position to False when used.
True if any positions remain uncleared (unused).
def go(x):
xs = bools.copy()
while x:
xs[x % base] = False
x //= base
return any(xs)
return lambda n: go(n * n)
- digit :: Int -> Char
def digit(n):
Digit character for given integer. return '0123456789abcdef'[n]
- TEST ----------------------------------------------------
- main :: IO ()
def main():
Smallest perfect squares using all digits in bases 2-16
start = time()
print(main.__doc__ + ':\n\nBase Root Square')
q = 0
for b in enumFromTo(2)(16):
q = allDigitSquare(b, q)
print(
str(b).rjust(2, ' ') + ' -> ' +
showIntAtBase(b)(digit)(q)().rjust(8, ' ') + ' -> ' +
showIntAtBase(b)(digit)(q * q)()
)
print(
'\nc. ' + str(ceil(time() - start)) + ' seconds.'
)
- GENERIC -------------------------------------------------
- enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
Integer enumeration from m to n. return lambda n: list(range(m, 1 + n))
- showIntAtBase :: Int -> (Int -> String) -> Int -> String -> String
def showIntAtBase(base):
String representation of an integer in a given base,
using a supplied function for the string representation
of digits.
def wrap(toChr, n, rs):
def go(nd, r):
n, d = nd
r_ = toChr(d) + r
return go(divmod(n, base), r_) if 0 != n else r_
return 'unsupported base' if 1 >= base else (
'negative number' if 0 > n else (
go(divmod(n, base), rs))
)
return lambda toChr: lambda n: lambda rs: (
wrap(toChr, n, rs)
)
- MAIN ---
if __name__ == '__main__':
main()</lang>
- Output:
Smallest perfect squares using all digits in bases 2-16: Base Root Square 2 -> 10 -> 100 3 -> 22 -> 2101 4 -> 33 -> 3201 5 -> 243 -> 132304 6 -> 523 -> 452013 7 -> 1431 -> 2450361 8 -> 3344 -> 13675420 9 -> 11642 -> 136802574 10 -> 32043 -> 1026753849 11 -> 111453 -> 1240a536789 12 -> 3966b9 -> 124a7b538609 13 -> 3828943 -> 10254773ca86b9 14 -> 3a9db7c -> 10269b8c57d3a4 15 -> 1012b857 -> 102597bace836d4 16 -> 404a9d9b -> 1025648cfea37bd9 c. 30 seconds.
REXX
The REXX language doesn't have
a sqrt function, nor does it have a general purpose
radix (base) convertor,
so RYO versions were included here.
These REXX versions can handle up to base 36.
slightly optimized
<lang rexx>/*REXX program finds/displays the first perfect square with N unique digits in base N.*/ numeric digits 40 /*ensure enough decimal digits for a #.*/ parse arg n . /*obtain optional argument from the CL.*/ if n== | n=="," then n= 16 /*not specified? Then use the default.*/ @start= 1023456789abcdefghijklmnopqrstuvwxyz /*contains the start # (up to base 36).*/
w= length(n) /* [↓] find the smallest square with */
do j=2 to n; beg= left(@start, j) /* N unique digits in base N. */
do k=iSqrt( base(beg,10,j) ) until #==0 /*start each search from smallest sqrt.*/
$= base(k*k, j, 10) /*calculate square, convert to base J. */
$u= $; upper $u /*get an uppercase version fast count. */
#= verify(beg, $u) /*count differences between 2 numbers. */
end /*k*/
say 'base' right(j,w) " root=" right(base(k,j,10),max(5,n)) ' square=' $
end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ base: procedure; arg x 1 #,toB,inB /*obtain: three arguments. */
@l= '0123456789abcdefghijklmnopqrstuvwxyz' /*lowercase (Latin or English) alphabet*/
@u= @l; upper @u /*uppercase " " " " */
if inb\==10 then /*only convert if not base 10. */
do; #= 0 /*result of converted X (in base 10).*/
do j=1 for length(x) /*convert X: base inB ──► base 10. */
#= # * inB + pos(substr(x,j,1), @u)-1 /*build a new number, digit by digit. */
end /*j*/ /* [↑] this also verifies digits. */
end
y= /*the value of X in base B (so far).*/
if tob==10 then return # /*if TOB is ten, then simply return #.*/
do while # >= toB /*convert #: base 10 ──► base toB.*/
y= substr(@l, (# // toB) + 1, 1)y /*construct the output number. */
#= # % toB /* ··· and whittle # down also. */
end /*while*/ /* [↑] algorithm may leave a residual.*/
return substr(@l, # + 1, 1)y /*prepend the residual, if any. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: procedure; parse arg x; r=0; q=1; do while q<=x; q=q*4; end
do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end; return r</lang>
- output when using the default input:
base 2 root= 10 square= 100 base 3 root= 22 square= 2101 base 4 root= 33 square= 3201 base 5 root= 243 square= 132304 base 6 root= 523 square= 452013 base 7 root= 1431 square= 2450361 base 8 root= 3344 square= 13675420 base 9 root= 11642 square= 136802574 base 10 root= 32043 square= 1026753849 base 11 root= 111453 square= 1240a536789 base 12 root= 3966b9 square= 124a7b538609 base 13 root= 3828943 square= 10254773ca86b9 base 14 root= 3a9db7c square= 10269b8c57d3a4 base 15 root= 1012b857 square= 102597bace836d4 base 16 root= 404a9d9b square= 1025648cfea37bd9
more optimized
This REXX version uses a highly optimized base function since it was that particular function that was consuming the majority of the CPU time.
It is about 10% faster. <lang rexx>/*REXX program finds/displays the first perfect square with N unique digits in base N.*/ numeric digits 40 /*ensure enough decimal digits for a #.*/ parse arg n . /*obtain optional argument from the CL.*/ if n== | n=="," then n= 16 /*not specified? Then use the default.*/ @start= 1023456789abcdefghijklmnopqrstuvwxyz /*contains the start # (up to base 36).*/ call base /*initialize 2 arrays for BASE function*/
/* [↓] find the smallest square with */
do j=2 to n; beg= left(@start, j) /* N unique digits in base N. */
do k=iSqrt( base(beg,10,j) ) until #==0 /*start each search from smallest sqrt.*/
$= base(k*k, j, 10) /*calculate square, convert to base J. */
#= verify(beg, $) /*count differences between 2 numbers. */
end /*k*/
say 'base' right(j, length(n) ) " root=" ,
lower( right( base(k, j, 10), max(5, n) ) ) ' square=' lower($)
end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ base: procedure expose !. !!.; arg x 1 #,toB,inB /*obtain: three arguments. */
@= 0123456789abcdefghijklmnopqrstuvwxyz /*the characters for the Latin alphabet*/
if x== then do i=1 for length(@); _= substr(@, i, 1); m= i - 1; !._= m
!!.m= substr(@, i, 1)
if i==length(@) then return /*Done with shortcuts? Then go back. */
end /*i*/ /* [↑] assign shortcut radix values. */
if inb\==10 then /*only convert if not base 10. */
do; #= 0 /*result of converted X (in base 10).*/
do j=1 for length(x) /*convert X: base inB ──► base 10. */
_= substr(x, j, 1); #= # * inB + !._ /*build a new number, digit by digit. */
end /*j*/ /* [↑] this also verifies digits. */
end
y= /*the value of X in base B (so far).*/
if tob==10 then return # /*if TOB is ten, then simply return #.*/
do while # >= toB /*convert #: base 10 ──► base toB.*/
_= # // toB; y= !!._ || y /*construct the output number. */
#= # % toB /* ··· and whittle # down also. */
end /*while*/ /* [↑] algorithm may leave a residual.*/
return !!.# || y /*prepend the residual, if any. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: procedure; parse arg x; r=0; q=1; do while q<=x; q=q*4; end
do while q>1; q=q%4; _=x-r-q; r=r%2; if _>=0 then do;x=_;r=r+q; end; end; return r
/*──────────────────────────────────────────────────────────────────────────────────────*/ lower: @abc= 'abcdefghijklmnopqrstuvwxyz'; return translate(arg(1), @abc, translate(@abc))</lang>
- output is identical to the 1st REXX version.
Sidef
<lang ruby>func first_square(b) {
var start = [1, 0, (2..^b)...].flip.map_kv{|k,v| v * b**k }.sum.isqrt
start..Inf -> first_by {|k|
k.sqr.digits(b).freq.len == b
}.sqr
}
for b in (2..16) {
var s = first_square(b)
printf("Base %2d: %10s² == %s\n", b, s.isqrt.base(b), s.base(b))
}</lang>
- Output:
Base 2: 10² == 100 Base 3: 22² == 2101 Base 4: 33² == 3201 Base 5: 243² == 132304 Base 6: 523² == 452013 Base 7: 1431² == 2450361 Base 8: 3344² == 13675420 Base 9: 11642² == 136802574 Base 10: 32043² == 1026753849 Base 11: 111453² == 1240a536789 Base 12: 3966b9² == 124a7b538609 Base 13: 3828943² == 10254773ca86b9 Base 14: 3a9db7c² == 10269b8c57d3a4 Base 15: 1012b857² == 102597bace836d4 Base 16: 404a9d9b² == 1025648cfea37bd9
Visual Basic .NET
This is faster than the Go version, but not as fast as the Pascal version. The Pascal version uses an array of integers to represent the square, as it's more efficient to increment and check that way.
This Visual Basic .NET version uses BigInteger variables for computation. It's quick enough for up to base19, tho.<lang vbnet>Imports System.Numerics
Module Program
Dim base, bm1 As Byte, hs As New HashSet(Of Byte), st0 As DateTime Const chars As String = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz|"
' converts base10 to string, using current base
Function toStr(ByVal b As BigInteger) As String
toStr = "" : Dim re As BigInteger : While b > 0
b = BigInteger.DivRem(b, base, re) : toStr = chars(CByte(re)) & toStr : End While
End Function
' checks for all digits present, checks every one (use when extra digit is present)
Function allIn(ByVal b As BigInteger) As Boolean
Dim re As BigInteger : hs.Clear() : While b > 0 : b = BigInteger.DivRem(b, base, re)
hs.Add(CByte(re)) : End While : Return hs.Count = base
End Function
' checks for all digits present, bailing when duplicates occur (can't use when extra digit is present)
Function allInQ(ByVal b As BigInteger) As Boolean
Dim re As BigInteger, c As Integer = 0 : hs.Clear() : While b > 0 : b = BigInteger.DivRem(b, base, re)
hs.Add(CByte(re)) : c += 1 : If c <> hs.Count Then Return False
End While : Return True
End Function
' converts string to base 10, using current base
Function to10(s As String) As BigInteger
to10 = 0 : For Each i As Char In s : to10 = to10 * base + chars.IndexOf(i) : Next
End Function
' returns minimum string representation, optionally inserting a digit
Function fixup(n As Integer) As String
fixup = chars.Substring(0, base)
If n > 0 Then fixup = fixup.Insert(n, n.ToString)
fixup = "10" & fixup.Substring(2)
End Function
' returns close approx.
Function IntSqRoot(v As BigInteger) As BigInteger
IntSqRoot = New BigInteger(Math.Sqrt(CDbl(v))) : Dim term As BigInteger
Do : term = v / IntSqRoot : If BigInteger.Abs(term - IntSqRoot) < 2 Then Exit Do
IntSqRoot = (IntSqRoot + term) / 2 : Loop Until False
End Function
' tabulates one base
Sub doOne()
bm1 = base - 1 : Dim dr As Byte = 0 : If (base And 1) = 1 Then dr = base >> 1
Dim id As Integer = 0, inc As Integer = 1, i As Long = 0, st As DateTime = DateTime.Now
Dim sdr(bm1 - 1) As Byte, rc As Byte = 0 : For i = 0 To bm1 - 1 : sdr(i) = (i * i) Mod bm1
rc += If(sdr(i) = dr, 1, 0) : sdr(i) += If(sdr(i) = 0, bm1, 0) : Next : i = 0
If dr > 0 Then
id = base : For i = 1 To dr : If sdr(i) >= dr Then If id > sdr(i) Then id = sdr(i)
Next : id -= dr : i = 0 : End If
Dim sq As BigInteger = to10(fixup(id)), rt As BigInteger = IntSqRoot(sq) + 0,
dn As BigInteger = (rt << 1) + 1, d As BigInteger = 1
sq = rt * rt : If base > 3 AndAlso rc > 0 Then
While sq Mod bm1 <> dr : rt += 1 : sq += dn : dn += 2 : End While ' alligns sq to dr
inc = bm1 \ rc : If inc > 1 Then dn += rt * (inc - 2) - 1 : d = inc * inc
dn += dn + d
End If : d <<= 1 : If base > 5 AndAlso rc > 0 Then : Do : If allInQ(sq) Then Exit Do
sq += dn : dn += d : i += 1 : Loop Until False : Else : Do : If allIn(sq) Then Exit Do
sq += dn : dn += d : i += 1 : Loop Until False : End If : rt += i * inc
Console.WriteLine("{0,3} {1,3} {2,2} {3,20} -> {4,-38} {5,10} {6,8:0.000}s {7,8:0.000}s",
base, inc, If(id = 0, " ", id.ToString), toStr(rt), toStr(sq), i,
(DateTime.Now - st).TotalSeconds, (DateTime.Now - st0).TotalSeconds)
End Sub
Sub Main(args As String())
st0 = DateTime.Now
Console.WriteLine("base inc id root square" & _
" test count time total")
For base = 2 To 28 : doOne() : Next
Console.WriteLine("Elasped time was {0,8:0.00} minutes", (DateTime.Now - st0).TotalMinutes)
End Sub
End Module</lang>
- Output:
This output is on a somewhat modern PC. For comparison, it takes TIO.run around 30 seconds to reach base20, so TIO.run is around 3 times slower there.
base inc id root square test count time total 2 1 10 -> 100 1 0.007s 0.057s 3 1 22 -> 2101 5 0.000s 0.057s 4 3 33 -> 3201 2 0.001s 0.058s 5 1 2 243 -> 132304 15 0.000s 0.058s 6 5 523 -> 452013 20 0.000s 0.059s 7 6 1431 -> 2450361 35 0.000s 0.059s 8 7 3344 -> 13675420 41 0.000s 0.059s 9 4 11642 -> 136802574 289 0.000s 0.059s 10 3 32043 -> 1026753849 17 0.000s 0.059s 11 10 111453 -> 1240A536789 1498 0.001s 0.060s 12 11 3966B9 -> 124A7B538609 6883 0.005s 0.065s 13 1 3 3828943 -> 10254773CA86B9 8243 0.013s 0.078s 14 13 3A9DB7C -> 10269B8C57D3A4 1330 0.000s 0.078s 15 14 1012B857 -> 102597BACE836D4 4216 0.003s 0.081s 16 15 404A9D9B -> 1025648CFEA37BD9 18457 0.012s 0.093s 17 1 1 423F82GA9 -> 101246A89CGFB357ED 195113 0.341s 0.434s 18 17 44B482CAD -> 10236B5F8EG4AD9CH7 30440 0.022s 0.456s 19 6 1011B55E9A -> 10234DHBG7CI8F6A9E5 93021 0.068s 0.524s 20 19 49DGIH5D3G -> 1024E7CDI3HB695FJA8G 11310604 8.637s 9.162s 21 1 6 4C9HE5FE27F -> 1023457DG9HI8J6B6KCEAF 601844 1.181s 10.342s 22 21 4F94788GJ0F -> 102369FBGDEJ48CHI7LKA5 27804949 21.677s 32.020s 23 22 1011D3EL56MC -> 10234ACEDKG9HM8FBJIL756 17710217 14.292s 46.312s 24 23 4LJ0HDGF0HD3 -> 102345B87HFECKJNIGMDLA69 4266555 3.558s 49.871s 25 12 1011E145FHGHM -> 102345DOECKJ6GFB8LIAM7NH9 78092125 69.914s 119.785s 26 5 52K8N53BDM99K -> 1023458LO6IEMKG79FPCHNJDBA 402922569 365.929s 485.714s 27 26 1011F11E37OBJJ -> 1023458ELOMDHBIJFGKP7CQ9N6A 457555293 420.607s 906.321s 28 9 58A3CKP3N4CQD7 -> 1023456CGJBIRQEDHP98KMOAN7FL 749593055 711.660s 1617.981s Elasped time was 26.97 minutes
Base29 seems to take an order of magnitude longer. I'm looking into some shortcuts.
zkl
<lang zkl>fcn squareSearch(B){
basenumerals:=B.pump(String,T("toString",B)); // 13 --> "0123456789abc"
highest:=("10"+basenumerals[2,*]).toInt(B); // 13 --> "10" "23456789abc"
foreach n in ([highest.toFloat().sqrt().toInt() .. highest]){
ns:=(n*n).toString(B);
if(""==(basenumerals - ns) ) return(n.toString(B),ns);
}
Void
}</lang> <lang zkl>println("Base Root N"); foreach b in ([2..16])
{ println("%2d %10s %s".fmt(b,squareSearch(b).xplode())) }</lang>
- Output:
Base Root N 2 10 100 3 22 2101 4 33 3201 5 243 132304 6 523 452013 7 1431 2450361 8 3344 13675420 9 11642 136802574 10 32043 1026753849 11 111453 1240a536789 12 3966b9 124a7b538609 13 3828943 10254773ca86b9 14 3a9db7c 10269b8c57d3a4 15 1012b857 102597bace836d4 16 404a9d9b 1025648cfea37bd9