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A065906 Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2. 5

%I

%S 15,48,55,197,206,221,235,283,297,408,444,472,489,577,578,623,641,677,

%T 701,703,763,854,930,1049,1081,1134,1140,1159,1160,1201,1253,1303,

%U 1311,1328,1374,1385,1415,1458,1459,1495,1501,1517,1557,1585,1714,1723,1726

%N Integers i > 1 for which there are three primes p such that i is a solution mod p of x^4 = 2.

%C Solutions mod p are represented by integers from 0 to p-1. The following equivalences holds for i > 1: There is a prime p such that i is a solution mod p of x^4 = 2 iff i^4 - 2 has a prime factor > i; i is a solution mod p of x^4 = 2 iff p is a prime factor of i^4 - 2 and p > i. i^4 - 2 has at most three prime factors > i. For i such that i^4 - 2 has no resp. one resp. two prime factors > i cf. A065903 resp. A065904 resp. A065905.

%F a(n) = n-th integer i such that i^4 - 2 has three prime factors > i.

%e a(3) = 55, since 55 is (after 15 and 48) the third integer i for which there are three primes p > i (viz. 73, 103 and 1217) such that i is a solution mod p of x^4 = 2, or equivalently, 55^4 - 2 = 9150623 = 73*103*1217 has three prime factors > 4. (cf. A065902).

%o (PARI): a065906(m) = local(c,n,f,a,s,j); c = 0; n = 2; while(c<m,f = factor(n^4-2); a = matsize(f)[1]; s = []; for(j = 1,a, if(f[j,1]>n,s = concat(s,f[j,1]))); if(matsize(s)[2] == 3,print1(n,","); c++); n++) a065906(50)

%Y Cf. A040028, A065902, A065903, A065904, A065905.

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Nov 28 2001

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Last modified August 3 07:47 EDT 2021. Contains 346435 sequences. (Running on oeis4.)