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

1. 
   
2. #include <gmp.h>
   
3. #include <stdio.h>
   
4. #include <stdlib.h>
   
5. 
   
6. mpz_t t;
   
7. unsigned int d[3] = {1, 4, 9}, maxlvl;
   
8. mpz_t l[32];
   
9. 
   
10. int check(mpz_t n)
    
11. {
    
12. //    gmp_printf("%Zd\n", n);
    
13. 
    
14.     if (mpz_perfect_square_p(n))
    
15.     {
    
16.        mpz_sqrt(t, n);
    
17.        gmp_printf("[%Zd, %Zd]\n", n, t);
    
18.     }     
    
19. }
    
20. 
    
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.   }
    
51. 
    
52.   for (i = 0; i < 32; i ++)
    
53.     mpz_clear(l[i]);
    
54.   mpz_clear(t);
    
55.   return 0;
    
56. }
    

复制代码

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