1,1

For all primes p>2 and integers gcd(x,y,p)=1, x^(p-1)/2 +- y^(p-1)/2 is divisible by p. This is so because (x^(p-1)/2 - y^(p-1)/2)(x^(p-1)/2 + y^(p-1)/2) = N = x^(p-1) - y^(p-1). Now N is divisible by p for x,y, (xy,p)=1 from Fermat's Little Theorem (flt) prime p divides a^(p-1) - 1 for all a, (a,p) = 1. Then X = x^(p-1)/2 - 1 is divisible by p and Y = y^(p-1)/2 - 1 is divisible by p. This implies X-Y is divisible by p and hence N is divisible by p. N is a general case of flt. Proving N is divisible by p not using flt will serve to prove flt as a special case of N with y = 1. For N, p=2 is allowed only if x and y have the same parity.

Primes p congruent to 1 or 11 modulo 12. - Michael Somos Aug 28 2006

Primes p such that 3 is a quadratic residue modulo p. - Michael Somos Aug 28 2006

p=3 is excluded by definition

For p=5, 3^2 - 1 = 8 <> 3k for any integer k, So 5 is not in the table.

For p=11, 3^5 - 1 = 242 = 11*22, so 11 is in the table.

(PARI) \s = +-1, d=diff ptopm1d2(n, x, d, s) = { forprime(p=3, n, p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0, print1(p", "))) }

(PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( c<n, m++; if( isprime(m)& kronecker(3, m)==1, c++)); m)} /* Michael Somos Aug 28 2006 */

Sequence in context: A136058 A106073 A072330 this_sequence A127043 A084952 A090433

Adjacent sequences: A097930 A097931 A097932 this_sequence A097934 A097935 A097936

nonn

Cino Hilliard (hillcino368(AT)gmail.com), Sep 04 2004