Talk:Brazilian numbers: Difference between revisions
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(some observations hopefully to limit the need of tests) |
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slots: base/number
<=1/12 <= 2/12 <=3/12 <=4/12
30717 4013 2225 3473 0 0 0 0 0 0 0 0
</pre>
:: 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. -- [[User:Gerard Schildberger|Gerard Schildberger]] ([[User talk:Gerard Schildberger|talk]]) 21:51, 15 August 2019 (UTC)
::: some more observations by factorization of the numbers:<BR>Brazilian primes always have "1" as digit.MaxBase = trunc(sqrt(prime))-> "111" and therefor are rare 213 out of 86400.<BR>So one need only to test if digit is "1" for prime numbers.
<pre>
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</pre>
How about nonprime odd numbers?
<pre>
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</pre>
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.<Br><Br>
Edit.Some more investigation:<BR>Which numbers are nonbrazilian :-)<Br>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
<pre>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</pre>
[[user:Horst.h|Horst.h]]
== Maple&Pari ==
Code can be seen at: [http://oeis.org/A125134 A125134] [[User:Billymacc|Billymacc]] ([[User talk:Billymacc|talk]]) 21:23, 23 July 2022 (UTC)
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