A390617 Primes p such that the norm of Mersenne prime M_{p, 2 + phi} = (2 + phi)^p - 1 in quadratic ring Z[phi], with phi = (1 + sqrt(5))/2, is a rational prime.

2, 7, 11, 17, 19, 23, 53, 67, 139, 173, 757, 1447, 1481, 1511, 1933, 2417

Offset

1,1

Comments

The norms of the corresponding quadratic Mersenne primes are in A390618.

Generalized Mersenne primes are defined M_{p, base} = (base^p - 1) / (base - 1). The division is redundant here, since base 2 + phi equals a unit + 1 in real quadratic field K = Q(sqrt(5)) and the norm remains unchanged.

2 + phi has least prime norm 5 in K. Base sqrt(5) =-1 + 2*phi with opposite norm -5 gives the related sequence A004061 (primes p such that (5^p - 1)/4 is prime), using definition (base^p + 1) / (base + 1) (p odd) because the norm is negative. A similar relation exists between prime exponents A323697 with base 2 + sqrt(2) and the Mersenne exponents A000043 with base sqrt(2), having norms 2 and -2 in Q(sqrt(2)). (Palimar and Shankar, p. 3; Smallbone p. 37)

Although the initial terms are regularly distributed, a gap of unknown size follows exponent 2417. 'Experimental evidence shows that Mersenne primes are sparse in Q(sqrt(d)) for d == 1 (mod 4).' (Palimar and Shankar, p. 6)

Links

Table of n, a(n) for n=1..16.

Sushma Palimar and B. R. Shankar, Mersenne primes in real quadratic fields, Journal of Integer Sequences, Vol. 15 (2012), #12.5.6.

Kieran Smallbone, Perfect numbers over simple algebraic number fields, PhD thesis, University of Oxford (2002).

Example

7 is in the sequence since (2 + phi)^7 - 1 = (8123 + 3625*sqrt(5))/2 and its norm 70001 = (8123^2 - 5*3625^2)/4 is prime.

Mathematica

Select[Prime[Range[360]], PrimeQ[AlgebraicNumberNorm[(GoldenRatio + 2)^# - 1]] &] (* Amiram Eldar, Nov 13 2025 *)

Prog

(PARI) {my(w = quadgen(5, 'w)); forprime(p=2, 1e4, if(ispseudoprime(norm((2+w)^p-1)), print1(p, ", ")))}

Crossrefs

Norms of the corresponding Mersenne primes: A390618.

Exponents M_{p,sqrt(5)}, with base norm -5: A004061.

Cf. A000043, A323697.

Sequence in context: A166005 A243009 A155009 * A384918 A045371 A380906

Adjacent sequences: A390614 A390615 A390616 * A390618 A390619 A390620

Keyword

nonn,hard,more

Author

Sjoerd J. Schaper, Nov 13 2025

Status

approved