Find palindromic numbers in both binary and ternary bases
- The task
Find and show (in decimal) the first six numbers (non-negative integers) that are palindromes in both base 2 and base 3.
Use zero (0) as the first number found, even though some other definitions ignore it.
Optionally, show the decimal number found in its binary and ternary form.
It's permissible to assume the first two numbers and simply list them.
- See also
- Sequence A60792, Numbers that are palindromic in bases 2 and 3 on The On-Line Encyclopedia of Integer Sequences.
C
Per the observations made by the Ruby code (which are correct), the numbers must have odd number of digits in base 3 with a 1 at the middle, and must have odd number of digits in base 2. <lang c>#include <stdio.h> typedef unsigned long long xint;
int is_palin2(xint n) { xint x = 0; if (!(n&1)) return !n; while (x < n) x = x<<1 | (n&1), n >>= 1; return n == x || n == x>>1; }
xint reverse3(xint n) { xint x = 0; while (n) x = x*3 + (n%3), n /= 3; return x; }
void print(xint n, xint base) { putchar(' '); // printing digits backwards, but hey, it's a palindrome do { putchar('0' + (n%base)), n /= base; } while(n); printf("(%lld)", base); }
void show(xint n) { printf("%llu", n); print(n, 2); print(n, 3); putchar('\n'); }
xint min(xint a, xint b) { return a < b ? a : b; } xint max(xint a, xint b) { return a > b ? a : b; }
int main(void) { xint lo, hi, lo2, hi2, lo3, hi3, pow2, pow3, i, n; int cnt;
show(0); cnt = 1;
lo = 0; hi = pow2 = pow3 = 1;
while (1) { for (i = lo; i < hi; i++) { n = (i * 3 + 1) * pow3 + reverse3(i); if (!is_palin2(n)) continue; show(n); if (++cnt >= 7) return 0; }
if (i == pow3) pow3 *= 3; else pow2 *= 4;
while (1) { while (pow2 <= pow3) pow2 *= 4;
lo2 = (pow2 / pow3 - 1) / 3; hi2 = (pow2 * 2 / pow3 - 1) / 3 + 1; lo3 = pow3 / 3; hi3 = pow3;
if (lo2 >= hi3) pow3 *= 3; else if (lo3 >= hi2) pow2 *= 4; else { lo = max(lo2, lo3); hi = min(hi2, hi3); break; } } } return 0; }</lang>
- Output:
0 0(2) 0(3) 1 1(2) 1(3) 6643 1100111110011(2) 100010001(3) 1422773 101011011010110110101(2) 2200021200022(3) 5415589 10100101010001010100101(2) 101012010210101(3) 90396755477 1010100001100000100010000011000010101(2) 22122022220102222022122(3) 381920985378904469 10101001100110110110001110011011001110001101101100110010101(2) 2112200222001222121212221002220022112(3)
D
<lang d>import core.stdc.stdio;
bool isPalindrome2(ulong n) pure nothrow @nogc {
ulong x = 0;
if (!(n & 1))
return !n;
while (x < n) {
x = (x << 1) | (n & 1);
n >>= 1;
}
return n == x || n == (x >> 1);
}
ulong reverse3(ulong n) pure nothrow @nogc {
ulong x = 0;
while (n) {
x = x * 3 + (n % 3);
n /= 3;
}
return x;
}
void printReversed(ubyte base)(ulong n) @nogc {
putchar(' ');
do {
putchar('0' + (n % base));
n /= base;
} while(n);
printf("(%d)", base);
}
void main() @nogc {
ulong i, cnt, top = 1, mul = 1, even = 0; uint count = 0;
for (i = cnt = 0; cnt < 6; i++) {
if (i == top) {
if (even ^= 1)
top *= 3;
else {
i = mul;
mul = top;
}
}
immutable n = i * mul + reverse3(even ? i / 3 : i);
if (isPalindrome2(n)) {
printf("%llu", n);
printReversed!3(n);
printReversed!2(n);
putchar('\n');
if (++count >= 6) // Print first 6.
break;
}
}
}</lang>
- Output:
0 0(3) 0(2) 1 1(3) 1(2) 6643 100010001(3) 1100111110011(2) 1422773 2200021200022(3) 101011011010110110101(2) 5415589 101012010210101(3) 10100101010001010100101(2) 90396755477 22122022220102222022122(3) 1010100001100000100010000011000010101(2)
Java
This takes a while to get to the 6th one (I didn't time it precisely, but it was less than 2 hours on an i7) <lang java>public class Pali23 { public static boolean isPali(String x){ return x.equals(new StringBuilder(x).reverse().toString()); }
public static void main(String[] args){
for(long i = 0, count = 0; count < 6;i++){ if((i & 1) == 0 && (i != 0)) continue; //skip non-zero evens, nothing that ends in 0 in binary can be in this sequence //maybe speed things up through short-circuit evaluation by putting toString in the if //testing up to 10M, base 2 has slightly fewer palindromes so do that one first if(isPali(Long.toBinaryString(i)) && isPali(Long.toString(i, 3))){ System.out.println(i + ", " + Long.toBinaryString(i) + ", " + Long.toString(i, 3)); count++; } } } }</lang>
- Output:
0, 0, 0 1, 1, 1 6643, 1100111110011, 100010001 1422773, 101011011010110110101, 2200021200022 5415589, 10100101010001010100101, 101012010210101 90396755477, 1010100001100000100010000011000010101, 22122022220102222022122
PARI/GP
<lang parigp>check(n)={ \\ Check for 2n+1-digit palindromes in base 3
my(N=3^n);
forstep(i=N+1,2*N,[1,2],
my(base2,base3=digits(i,3),k);
base3=concat(Vecrev(base3[2..n+1]), base3);
k=subst(Pol(base3),'x,3);
base2=binary(k);
if(base2==Vecrev(base2), print1(", "k))
)
}; print1("0, 1"); for(i=1,11,check(i))</lang>
- Output:
0, 1, 6643, 1422773, 5415589, 90396755477
Perl 6
Instead of searching for numbers that are palindromes in one base then checking the other, generate palindromic trinary numbers directly, then check to see if they are also binary palindromes. Outputs the list in decimal, binary and trinary.
Note: finding the first six takes a depressingly long time with this implementation. Seven would be unbearable. :-(
<lang perl6>my @palindromes := 0, 1, gather for 1 .. * -> $n {
my $p = $n.base(3);
my $q = $p.flip;
for "$p$q", "{$p}0$q", "{$p}1$q", "{$p}2$q" -> $r {
my $s = :3($r);
next if $s %% 2;
$s = $s.base(2);
next unless $s eq $s.flip;
take :2($s);
}
}
printf "%d, %s, %s\n", $_, $_.base(2), $_.base(3) for @palindromes[^6]</lang>
- Output:
0, 0, 0 1, 1, 1 6643, 1100111110011, 100010001 1422773, 101011011010110110101, 2200021200022 5415589, 10100101010001010100101, 101012010210101 90396755477, 1010100001100000100010000011000010101, 22122022220102222022122
Python
<lang python>from itertools import islice
digits = "0123456789abcdefghijklmnopqrstuvwxyz"
def baseN(num,b):
if num == 0: return "0" result = "" while num != 0: num, d = divmod(num, b) result += digits[d] return result[::-1] # reverse
def pal2(num):
if num == 0 or num == 1: return True based = bin(num)[2:] return based == based[::-1]
def pal_23():
yield 0
yield 1
n = 1
while True:
n += 1
b = baseN(n, 3)
revb = b[::-1]
#if len(b) > 12: break
for trial in ('{0}{1}'.format(b, revb), '{0}0{1}'.format(b, revb),
'{0}1{1}'.format(b, revb), '{0}2{1}'.format(b, revb)):
t = int(trial, 3)
if pal2(t):
yield t
for pal23 in islice(pal_23(), 6):
print(pal23, baseN(pal23, 3), baseN(pal23, 2))</lang>
- Output:
0 0 0 1 1 1 6643 100010001 1100111110011 1422773 2200021200022 101011011010110110101 5415589 101012010210101 10100101010001010100101 90396755477 22122022220102222022122 1010100001100000100010000011000010101
REXX
Programming note: This version is quite a bit faster than the previous one.
For this REXX program, a few deterministic assumptions were made:
- for the requirement of binary palindromes, the number of binary digits have to be odd.
- for the requirement of ternary palindromes, the numbers can't end in zero (in base 3).
The method used is to (not find, but) construct a binary palindrome by:
- using the binary version of a number (abcdef), which may end in binary zeroes,
- flipping the binary digits (fedcba),
- constructing two binary palindromes:
- abcdef || 0 || fedcba and
- abcdef || 1 || fedcba
- (the above 2 steps ensures an odd number of binary digits),
- ensure the decimal versions are not divisible by 3,
- convert the decimal numbers to base 3,
- check for palindromic numbers in base 3.
<lang rexx>/*REXX pgm finds decimal #s that are palindromic in binary and ternary.*/ numeric digits 50 /*biggest known B2B3 pal: 44 dig.*/ parse arg H .; if H== then H=6 /*when H hits are found, stop. */ hits=0 /*number of hits (palindromes). */ say right(0, digits()); hits=hits+1 /*display the 1st pal (in B2&B3).*/ say right(1, digits()); hits=hits+1 /*display the 2nd pal (in B2&B3).*/ !.= /* [↓] build list of powers of 3*/
do i=1 until pos('E',!.i)\==0; !.i=3**i; end /*i*/
o=1 /* [↓] primary search: bin pals.*/
do n=1 /*use all #s, however, DEC == odd*/ bin=strip(x2b(d2x(n)), 'L', 0) /*convert the number to binary. */ rev=reverse(bin) /*reverse the binary digits. */ dec=x2d(b2x(bin'0'rev)) /*construct a binary palindrom«0»*/ if dec//3\==0 then call rad3 /*if not divisible by 3, try it. */ dec=x2d(b2x(bin'1'rev)) /*construct a binary palindrom«1»*/ if dec//3\==0 then call rad3 /*if not divisible by 3, try it. */ end /*n*/ /* [↑] run until we find H hits.*/
/*──────────────────────────────────RAD3 subroutine─────────────────────*/ rad3: x=dec /*convert constructed # to base 3*/
do j=o; if !.j>x then leave; end /*find upper limit of pow of 3.*/
q= /*set base 3 result holder to nul*/
o=j-1 /*use this pow of 3 for next time*/
do k=o to 1 by -1; _=!.k; d=x%_; q=q||d; x=x//_; end /*k*/
t=q || x /*handle the conversion residual.*/
if t\==reverse(t) then return /*is this ternary # palindromic? */
say right(dec, digits()) /*display a palindrome (in B2&B3)*/
hits=hits+1 /*bump hit counter (palindromes).*/
if hits==H then exit /*stick a fork in it, we're done.*/
return /* [↑] this process is sluggish.*/</lang> output when using the input of: 7
0
1
6643
1422773
5415589
90396755477
381920985378904469
Ruby
This program is based on the fact that the double palindromic numbers in base 3 all have a "1" right in the middle. Also, both base 2 and base 3 representations have an odd number of digits.
- 1 digit under the number of the palindromic doesn't become zero.
- As for the N numbering-system, at the time of the multiple of N, 1 digit below becomes zero.
- Palindromic by the even-number digit binary system is 3 multiples.
- Palindromic by the even-number digit ternary-system is 4 multiples.
- In palindromic by the ternary-system of the odd digit, the value of the center position is an even number in case of "0" or "2".
This program constructs base 3 palindromes using the above "rules" and checks if they happen to be binary palindromes.
<lang ruby>pal23 = Enumerator.new do |y|
y << 0 y << 1 for i in 1 .. 1.0/0.0 # 1.step do |i| (Ruby 2.1+) n3 = i.to_s(3) n = (n3 + "1" + n3.reverse).to_i(3) n2 = n.to_s(2) y << n if n2.size.odd? and n2 == n2.reverse end
end
puts " decimal ternary binary" 6.times do |i|
n = pal23.next puts "%2d: %12d %s %s" % [i, n, n.to_s(3).center(25), n.to_s(2).center(39)]
end</lang>
- Output:
decimal ternary binary 0: 0 0 0 1: 1 1 1 2: 6643 100010001 1100111110011 3: 1422773 2200021200022 101011011010110110101 4: 5415589 101012010210101 10100101010001010100101 5: 90396755477 22122022220102222022122 1010100001100000100010000011000010101
zkl
Brute force, hits the wall hard after a(5) <lang zkl>fcn twoThreeP(n){
n2s:=n.toString(2); n3s:=n.toString(3); (n2s==n2s.reverse() and n3s==n3s.reverse())
} [0..].filter(5,twoThreeP) .apply2(fcn(n){println(n,"==",n.toString(3),"==",n.toString(2))}) </lang>
- Output:
0==0==0 1==1==1 6643==100010001==1100111110011 1422773==2200021200022==101011011010110110101 5415589==101012010210101==10100101010001010100101