p^2=a^2&b^2

medie2005

p^2=a^2&b^2 (primepuzzles: puzzle 483) I found that the following smaller primes having this property are: 7, 13, 19 & 41, because: 7^2=49=2^2&3^2 13^2= 169=4^2&3^2 19^2= 361=6^2&1^2 41^2= 1681=4^2&9^2 The largest one I found was this: 1012639687^2=1025439135687457969 = 320224786^2&3^2 Q1. Can you find larger examples? Q2 Can you find examples (other than 7^2=49=2^2&3^2) such that p, a & b are prime numbers? Q3: Is there a larger prime than 446653271 such that all the digits of p^2 are squares? 446653271^2= 199499144494999441 (Suggested model: p^2=a^2&b^2&c^2...z^2, a, b, ... z is 1 or 4 or 9) Q4. Find larger primes such that p^2=a^2&b^2&c^2...z^2 where a, b, ... z contains not necessarily just one digit as in Q3.

无心人

怀疑这个网站的建立目的是收集站长没能力计算的各种有趣的问题 让别人帮他算

无心人

Q3应该可以用Haskell暴力穷举 素数的开始数字是2，3，4

medie2005

谁来解Q4?

gxqcn

转帖网页 Squares containing at most three distinct digits 里部分结论： 3743127183788194652² = 14011001114014141144414101441441401104 43694278824566964251² = 1909190001999001011109190090109911991001 99704560597822753² = 9940999404004909449099404004499009 648070211589107021² = 419994999149149944149149944191494441 但很遗憾，这里面平方根均为合数。 这类特殊的完全平方数本来就稀缺，还要求平方根为素数，实在是大海捞针啊！

无心人

import ONeillPrimes(primesToLimit) valid n = all (\x -> (x == '1') || (x == '4') || (x == '9') ) \$ show n main = do let primes_set = primesToLimit 1000000000 let large_ps = filter (>= 40000000) primes_set let result = [(p, n) | p <- large_ps, let n = p^2, valid n] print \$ show result

gxqcn

如果把“0”算作完全平方数，我可能会找到符合要求的， 现在正在编程搜索。。。

无心人

Q3的那个数之前的： 1: [9, 3] 2: [49, 7] 5: [11449, 107] 然后是Q3的 18: [199499144494999441, 446653271]

无心人

2. #include <gmp.h>
3. #include <stdio.h>
4. #include <stdlib.h>
6. mpz_t t;
7. unsigned int d[3] = {1, 4, 9}, maxlvl;
8. mpz_t l[32];
10. int check(mpz_t n)
11. {
12. // gmp_printf("%Zd\n", n);
14. if (mpz_perfect_square_p(n))
15. {
16. mpz_sqrt(t, n);
17. gmp_printf("[%Zd, %Zd]\n", n, t);
18. }
19. }
21. int circle(int lvl)
22. {
23. if (lvl >= maxlvl)
24. {
25. check(l[lvl-1]);
26. }
27. else
28. for (int i = 0; i < 3; i ++ )
29. {
30. mpz_mul_ui(l[lvl], l[lvl-1], 10);
31. mpz_add_ui(l[lvl], l[lvl], d[i]);
32. circle(lvl + 1);
33. }
34. }
35. int main(void)
36. {
37. int i;
38. mpz_init(t);
39. for (i = 0; i < 32; i ++)
40. {
41. mpz_init(l[i]);
42. mpz_set_ui(l[i], 0);
43. }
44. printf("请输入数字长度：");
45. scanf("%u", &maxlvl);
46. for (i = 0; i < 3; i ++)
47. {
48. mpz_set_ui(l[0], d[i]);
49. circle(1);
50. }
52. for (i = 0; i < 32; i ++)
53. mpz_clear(l[i]);
54. mpz_clear(t);
55. return 0;
56. }

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是平方数的结果是凤毛麟角而已 勿论是素数平方了 所以干脆不判素性了

无心人

比如19位的唯一的 [9914419419914449449, 3148717107] 就不是素数