Talk:Brazilian numbers
wee discrepancy
Is it possible to be a little more specific regarding the "wee discrepancy" with the F# version? <lang fsharp> printfn "%d" (Seq.item 3999 (Brazilian())) </lang> prints 4618--Nigel Galloway (talk) 17:05, 14 August 2019 (UTC)
- OK I think I've found it--Nigel Galloway (talk) 17:37, 14 August 2019 (UTC)
- I also noticed the difference two days ago, and I assumed that my REXX version was incorrect and was trying to find what the problem was in my computer program; I was hoping somebody else would calculate the 100,000th Brazilian number and verify it (or not). -- Gerard Schildberger (talk) 19:23, 14 August 2019 (UTC)
some observations not proofs
I tried to check the maximal base needed for an odd brazilian number.
If a number is brazilian the maximal base to test is always less equal number / 3.
If a number is prime and brazilian then the maximal base is square root of number.
Try it online!
// only primes are shown
number base base*base
13 3 9
31 2 4
43 6 36
73 8 64
127 2 4
157 12 144
211 14 196
241 15 225
307 17 289
421 20 400
463 21 441
601 24 576
757 27 729
1093 3 9
1123 33 1089
1483 38 1444
..
55987 6 36
60271 245 60025
60763 246 60516
71023 266 70756
74257 272 73984
77563 278 77284
78121 279 77841
82657 287 82369
83233 288 82944
84391 290 84100
86143 293 85849
88741 17 289
95791 309 95481
98911 314 98596
odd brazilian numbers 7 .. 100000 : 40428
slots: base/number
<=1/12 <= 2/12 <=3/12 <=4/12
30717 4013 2225 3473 0 0 0 0 0 0 0 0
- Thanks, Mr. Horst (userid Horst.h), I added (the non-prime hint) to the REXX program and it speeded it up by a factor of two. -- Gerard Schildberger (talk) 21:51, 15 August 2019 (UTC)
- some more observations by factorization of the numbers:
Brazilian primes always have "1" as digit.MaxBase = trunc(sqrt(prime))-> "111" and therefor are rare 213 out of 86400.
So one need only to test if digit is "1" for prime numbers.
- some more observations by factorization of the numbers:
- Thanks, Mr. Horst (userid Horst.h), I added (the non-prime hint) to the REXX program and it speeded it up by a factor of two. -- Gerard Schildberger (talk) 21:51, 15 August 2019 (UTC)
number = factors base repeated digit
7 = 7 2 1 "111" to base 2
13 = 13 3 1 "111" to base 3
31 = 31 2 1 "11111" to base 2
43 = 43 6 1
73 = 73 8 1
127 = 127 2 1 "1111111" to base 2
157 = 157 12 1
--
601 = 601 24 1
757 = 757 27 1
1093 = 1093 3 1 "1111111" to base 3
...
987043 = 987043 993 1
1003003 = 1003003 1001 1
1005007 = 1005007 1002 1
1015057 = 1015057 1007 1
1023133 = 1023133 1011 1
1033273 = 1033273 1016 1
1041421 = 1041421 1020 1
1045507 = 1045507 1022 1
1059871 = 1059871 1029 1 "111" to base 1029
Max number 1084566 -> 84600 primes
Brazilian primes found 213
How about nonprime odd numbers?
number = factors base repeated digit
15 = 3*5 2 1 = "1111" also "33" to base 4 -> ( 5-1)
21 = 3*7 4 1 = "111" also "33" to base 6 -> ( 7-1)
27 = 3^3= 3*9 8 3
33 = 3*11 10 3
35 = 5*7 6 5
39 = 3*13 12 3
45 = 3^2*5 8 5
51 = 3*17 16 3
55 = 5*11 10 5
57 = 3*19 7 1 also "33" to base 18
63 = 3^2*7 2 1 also "77" to base 8
65 = 5*13 12 5
69 = 3*23 22 3
75 = 3*5^2 14 5
77 = 7*11 10 7
81 = 3^4=3*27 26 3
85 = 5*17 4 1 also "55" to base 16
87 = 3*29 28 3
91 = 7*13 9 1
93 = 3*31 5 3
95 = 5*19 18 5
99 = 3^2*11 10 9
105 = 3*5*7 14 7
111 = 3*37 10 1 also "33" to base 36
I think, taking the factorization of the number leave the highest factor -1 > sqrt( number) as base and the rest as digit.Something to test.Horst.h