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A097933
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Primes such that p divides 3^((p-1)/2) - 1.
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7
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11, 13, 23, 37, 47, 59, 61, 71, 73, 83, 97, 107, 109, 131, 157, 167, 179, 181, 191, 193, 227, 229, 239, 241, 251, 263, 277, 311, 313, 337, 347, 349, 359, 373, 383, 397, 409, 419, 421, 431, 433, 443, 457, 467, 479, 491, 503, 541, 563, 577, 587, 599, 601, 613
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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.
Apart from initial terms, this and A038874 are the same. - N. J. A. Sloane, May 31 2009
Primes in A091998. [Reinhard Zumkeller, Jan 07 2012]
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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EXAMPLE
| 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.
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PROG
| (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 */
(Haskell)
a097933 n = a097933_list !! (n-1)
a097933_list = [x | x <- a091998_list, a010051 x == 1]
-- Reinhard Zumkeller, Jan 07 2012
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CROSSREFS
| Cf. A038874.
Cf. A010051.
Sequence in context: A136058 A106073 A072330 * A166484 A127043 A084952
Adjacent sequences: A097930 A097931 A097932 * A097934 A097935 A097936
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KEYWORD
| nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Sep 04 2004
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