OFFSET
1,1
COMMENTS
The primes p of the corresponding Mersenne exponents are in A390617.
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 A086122 (primes of the form (5^p - 1)/4), using definition (base^p + 1) / (base + 1) (for p odd) because the norm is negative. A similar relation exists between primes A232993 with base 2 + sqrt(2) and the Mersenne primes A000668 with base sqrt(2), having norms 2 and -2 in Q(sqrt(2)). (Palimar and Shankar, p. 3; Smallbone p. 37)
Each M_{p, 2 + phi} norm has a base-5 expansion of length p, where the high-order digits are a string of approximately [p/5] fours. Division by 5^(p+1)/2 leaves remainder 1.
LINKS
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).
FORMULA
a(n) = norm((2 + phi)^A390617(n) - 1) in Q(sqrt(5)).
EXAMPLE
Prime 70001 is in the sequence since (2 + phi)^7 - 1 = (8123 + 3625*sqrt(5))/2 with norm (8123^2 - 5*3625^2)/4 = 70001 = 4220001_5.
MATHEMATICA
Select[AlgebraicNumberNorm[(GoldenRatio + 2)^Prime[Range[20]] - 1], PrimeQ] (* Amiram Eldar, Nov 13 2025 *)
PROG
(PARI) {my(w = quadgen(5, 'w)); forprime(p=2, 1e4, my(n=norm((2+w)^p-1)); if(ispseudoprime(n), print1(n, ", ")))}
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Sjoerd J. Schaper, Nov 13 2025
STATUS
approved