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First primes beginning a chain of 4 primes indexed equidistantly (n-th, (n+b)-th, (n+2b)-th, (n+3b)-th primes) whose sum of squares is the square of two times a prime and with b <= n.
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%I #8 Jul 10 2012 12:49:34

%S 137,199,223,773,2161,2477,3943,4079,4423,4603,6791,7297,7547,7559,

%T 12853,15299,17431,20807,22573,22637,25931,27179,31337,32027,32303,

%U 34403,36683,37573,38501,38671,41549,45523,48193,51941,57689,58679,60913,61471,61483

%N First primes beginning a chain of 4 primes indexed equidistantly (n-th, (n+b)-th, (n+2b)-th, (n+3b)-th primes) whose sum of squares is the square of two times a prime and with b <= n.

%C Note that 61471 and 61483 are consecutive primes; 65537 = 2^16+1 .The lowest indexed equidistance up to n=500000, is b=4 for n=46, p(46) = 199.

%e 199^2 + 229^2 + 251^2 + 271^2 = 478^2 and 239 is prime; b = 4.

%e 137^2 + 223^2 + 307^2 + 397^2 = 566^2 and 283 is prime; b = 15.

%e 223^2 + 307^2 + 397^2 + 487^2 = 734^2 and 367 is prime; b = 15.

%e 11^2 + 59^2 + 109^2 + 179^2 = 218^2 and 109 is prime; was not included because p(5) = 11 and b = 12 > 5.

%o (PARI) for(n=1,10000,for(b=1,n,a=(prime(n))^2+(prime(n+b))^2+(prime(n+2*b))^2+ (prime(n+3*b))^2;if(issquare(a)&isprime(sqrtint(a)/2),print1(prime(n)", "))))

%K nonn

%O 1,1

%A _Robin Garcia_, Jul 09 2012