A101411 -id:A101411 - cp's OEIS frontend

A118612 Denominator if the numerator and denominator of the continued fraction rational approximation of sqrt(2) are both prime.

2, 5, 29, 44560482149, 13558774610046711780701

Offset: 1

Joshua Zucker, May 08 2006

Next term, if it exists, is bigger than 489 digits (the 1279th convergent to sqrt(2)).

Andrej Dujella, Mirela Jukić Bokun, Ivan Soldo, A Pellian equation with primes and applications to D(-1)-quadruples, arXiv:1706.01959 [math.NT], 2017.

A086397 has the numerators. This sequence is a subsequence of A000129, A086383 and A101411.

For[n = 2, n < 1500, n++, a := Join[{1}, Table[2, {i, 2, n}]]; If[PrimeQ[Denominator[FromContinuedFraction[a]]], If[PrimeQ[Numerator[FromContinuedFraction[a]]], Print[Denominator[FromContinuedFraction[a]]]]]] (* Stefan Steinerberger, May 09 2006 *)

A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

Original entry on oeis.org

50, 1682, 3971273138702695316402, 367680737852094722224630791187352516632102802

Offset: 1

Rick L. Shepherd, Sep 03 2009

nonn

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.
Corresponding p are prime Pell numbers (prime denominators of continued fraction convergents to sqrt(2)).
Corresponding q are prime numerators of the continued fraction convergents to sqrt(2).
Corresponding p, q, p^2, q^2, (p,q), (q,p), etc., form subsequences of many other OEIS sequences; see cross-references.
Any further terms are too large to include here.

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

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.

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

cp's OEIS Frontend

A118612 Denominator if the numerator and denominator of the continued fraction rational approximation of sqrt(2) are both prime.

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