# Talk:Brazilian numbers

## wee discrepancy[edit]

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 100,000
^{th}Brazilian number and verify it (or not). -- Gerard Schildberger (talk) 19:23, 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 100,000

## some observations not proofs[edit]

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.

Edit.Some more investigation:

Which numbers are nonbrazilian :-)

As one can see, only primes are possibly nonbrazilian
and square numbers of odd primes are nonbrazilian with only one exception found up to 10000 : 11^2

factorization of the non brazilian numbers 9 = 3^2 11 = 11 17 = 17 19 = 19 23 = 23 25 = 5^2 29 = 29 37 = 37 41 = 41 47 = 47 49 = 7^2 53 = 53 59 = 59 61 = 61 67 = 67 71 = 71 79 = 79 83 = 83 89 = 89 97 = 97 101 = 101 103 = 103 107 = 107 109 = 109 113 = 113 131 = 131 137 = 137 139 = 139 149 = 149 151 = 151 163 = 163 167 = 167 169 = 13^2 173 = 173 179 = 179 181 = 181 191 = 191 193 = 193 197 = 197 199 = 199 223 = 223 227 = 227 229 = 229 233 = 233 239 = 239 251 = 251 257 = 257 263 = 263 269 = 269 271 = 271 277 = 277 281 = 281 283 = 283 289 = 17^2 293 = 293 311 = 311 313 = 313 317 = 317 331 = 331 337 = 337 347 = 347 349 = 349 353 = 353 359 = 359 361 = 19^2 367 = 367 373 = 373 379 = 379 383 = 383 389 = 389 397 = 397 401 = 401 409 = 409 419 = 419 431 = 431 433 = 433 439 = 439 443 = 443 449 = 449 457 = 457 461 = 461 467 = 467 479 = 479 487 = 487 491 = 491 499 = 499 503 = 503 509 = 509 521 = 521 523 = 523 529 = 23^2 541 = 541 547 = 547 557 = 557 563 = 563 569 = 569 571 = 571 577 = 577 587 = 587 593 = 593 599 = 599 607 = 607 613 = 613 617 = 617 619 = 619 631 = 631 641 = 641 643 = 643 647 = 647 653 = 653 659 = 659 661 = 661 673 = 673 677 = 677 683 = 683 691 = 691 701 = 701 709 = 709 719 = 719 727 = 727 733 = 733 739 = 739 743 = 743 751 = 751 761 = 761 769 = 769 773 = 773 787 = 787 797 = 797 809 = 809 811 = 811 821 = 821 823 = 823 827 = 827 829 = 829 839 = 839 841 = 29^2 853 = 853 857 = 857 859 = 859 863 = 863 877 = 877 881 = 881 883 = 883 887 = 887 907 = 907 911 = 911 919 = 919 929 = 929 937 = 937 941 = 941 947 = 947 953 = 953 961 = 31^2 967 = 967 971 = 971 977 = 977 983 = 983 991 = 991 997 = 997 Max number 1000 now checking sqr(primes) upto 10000: 121 = 11^2 last checked 9983^2 Brazilian found 1 99494 ms

## Maple&Pari[edit]

Code can be seen at: A125134 Billymacc (talk) 21:23, 23 July 2022 (UTC)