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Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.
0

%I #2 Mar 30 2012 17:36:45

%S 50,1682,3971273138702695316402,

%T 367680737852094722224630791187352516632102802

%N Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

%C A079704 INTERSECT A002522. Subsequence of A088920 (Solutions k to the Diophantine equation k = 2n^2 = m^2+1): those terms for which associated m in A002315 and n in A001653 are both prime.

%C Corresponding p are prime Pell numbers (prime denominators of continued fraction convergents to sqrt(2)).

%C Corresponding q are prime numerators of the continued fraction convergents to sqrt(2).

%C Corresponding p, q, p^2, q^2, (p,q), (q,p), etc., form subsequences of many other OEIS sequences; see cross-references.

%C Any further terms are too large to include here.

%F a(n) = 2*(A118612(n+1))^2 = (A086397(n+1))^2 + 1.

%e a(1) = 50 as 50 = 2*5^2 = 7^2 + 1, where 5 and 7 are prime.

%Y Cf. A088920, A118612, A086397, A086395, A002315 (NSW numbers), A088165 (prime NSW numbers = prime RMS numbers (A140480)), A001653, A000129 (Pell numbers), A086383, A101411, A079704, A002522, A008843, A104683, A163742, etc.

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Sep 03 2009