A118612 -id:A118612 - cp's OEIS frontend

A086397 Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.

3, 7, 41, 63018038201, 19175002942688032928599

Offset: 1

Cino Hilliard, Sep 06 2003

Next term, if it exists, is bigger than 489 digits (the 1279th convergent to sqrt(2)). - Joshua Zucker, May 08 2006
Are the terms >= 7 the primes in A183064? Is this a subsequence of A088165? - R. J. Mathar, Aug 16 2019
Yes, the terms >= 7 are the primes in A183064 and are a subsequence of A088165.  a(1)=3 is from the numerator of 3/2, but no other convergents > sqrt(2) can appear in this sequence because they all have even denominator.  All terms >= 7 are lower principal convergents from A002315/A088165/A183064 - Martin Fuller, Apr 08 2023

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

Denominators are A118612.

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[Numerator[FromContinuedFraction[a]]]]]] (* Stefan Steinerberger, May 09 2006 *)

cfracnumdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer)&&ispseudoprime(denom), print1(numer",");numer2=numer;denom2=denom); ) default(realprecision,28); }

More terms from Cino Hilliard, Jan 15 2005

Edited by N. J. A. Sloane, Aug 06 2009 at the suggestion of R. J. Mathar

A163742 Prime pairs (p,q) of the form p=A002315(k), q=A001653(k) for some k.

Original entry on oeis.org

7, 5, 41, 29, 63018038201, 44560482149, 19175002942688032928599, 13558774610046711780701

Offset: 1

Vincenzo Librandi, Aug 03 2009

By construction, all these pairs satisfy p^2 -2*q^2 = -1.

One can generate the combined A002315 and A001653 by a mixed recursion 3p+4q -> p; 2p+3q -> q, and then test after each step both p and q for primality.

			(A002315(1),A001653(1)) = (7,5) are both prime and define the first pair in the sequence.
(A002315(2),A001653(2)) = (41,29) are both prime and define the second pair in the sequence.
(A002315(3),A001653(3)) = (239,169=13^2) contain the composite 169 and do not contribute to the sequence.

V. Librandi, Pell's equation with prime solutions

Bisections are (essentially): A086397, A118612.

Definition clarified by R. J. Mathar, Aug 12 2009

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.

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A086397 Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.

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A163742 Prime pairs (p,q) of the form p=A002315(k), q=A001653(k) for some k.

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

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