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

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

Offset: 1

Author

Lekraj Beedassy, Apr 26 2005

Keywords

• hard
• more
• nonn

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

Examples

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

References

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

Links

• 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

Crossrefs

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

Programs

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

Extensions

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