A105318 Starting prime for the smallest prime Pythagorean sequence for n triangles.

5, 3, 271, 169219, 356498179, 2500282512131, 20594058719087111, 2185103796349763249

Offset

1,1

Comments

Smallest prime p(0) such that the n-chain governed by recurrence p(i+1)=(p(i)^2 + 1)/2 are all primes. Equivalently, least prime p(0) that generates a sequence of n 2-prime triangles, where p(k) is the hypotenuse of the k-th triangle and the leg of the (k+1)-th triangle.

For n>2, the last digit of a(n) is 1 or 9. - Ya-Ping Lu, May 17 2025

References

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 258.

Links

Table of n, a(n) for n=1..8.

C. K. Caldwell, The Prime Glossary, Pythagorean triples

H. Dubner, Posting to Number Theory List

H. Dubner and T. Forbes, Prime Pythagorean Triangles

H. Dubner and T. Forbes, Journal of Integer Sequences, Vol. 4(2001) #01.2.3, Prime Pythagorean triangles

T. Forbes, Posting to Number Theory List

T. Forbes, Posting to Number Theory List

Example

5 is a(1) because (5^2+1)/2 = 13 is prime, but (13^2+1)/2 = 85 is not.

Prog

(Python)
from sympy import isprime, nextprime; m = lambda x: (x*x+1)//2; p = 2; D = {}
while p < 2185103796349763249:
    p = nextprime(p); q = m(p); n = 1
    while isprime(q) and isprime(m(q)): n += 1; q = m(q)
    if n not in D: D.update({n: p})
[print(k, end =', ') for key, k in sorted(D.items())] # Ya-Ping Lu, May 17 2025

Crossrefs

Cf. A048161, A048270, A048295, A308635, A308636.

Sequence in context: A264738 A002208 A100653 * A304287 A121021 A237518

Adjacent sequences: A105315 A105316 A105317 * A105319 A105320 A105321

Keyword

hard,more,nonn

Author

Lekraj Beedassy, Apr 26 2005

Extensions

a(1) added by T. D. Noe, Jan 29 2011

Status

approved