A354155 Lagrange primes: primes p == 1 (mod 4) such that X = (p-1)/2 is the least solution in the interval [1, (p-1)/2] of the congruence (X!)^2 == -1 (mod p).

5, 13, 17, 29, 41, 53, 73, 97, 137, 229, 241, 257, 281, 313, 397, 409, 449, 457, 461, 521, 541, 617, 653, 661, 677, 733, 769, 809, 853, 857, 929, 953, 997, 1021, 1069, 1109, 1201, 1213, 1217, 1249, 1277, 1361, 1373, 1409, 1489, 1553, 1597, 1609, 1621, 1697

Offset: 1

Author

N. J. A. Sloane, May 29 2022, based on Section 18.5 of Cosgrave (2022)

Keywords

• nonn

References

• J. B. Cosgrave, A Mersenne-Wieferich Odyssey, Manuscript, May 2022. See Section 18.5.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..3100

Crossrefs

Cf. A002144, A354156.

Programs

PARI

is(n)=if(n%4 != 1 || !isprime(n), return(0)); my(t1=lift(sqrt(Mod(-1,n))), t2=n-t1, t=Mod(1,n)); for(k=2,n\2, if(t==t1 || t==t2, return(0)); t*=k); 1 \\ Charles R Greathouse IV, Aug 03 2023

PARI

list(lim)=my(v=List()); forprimestep(p=5,lim\1,4, my(t1=lift(sqrt(Mod(-1,p))),t2=p-t1, t=Mod(1,p)); for(k=2,p\2, if(t==t1 || t==t2, next(2)); t*=k); listput(v,p)); Vec(v) \\ Charles R Greathouse IV, Aug 03 2023

Python

from itertools import islice
from sympy import factorial, nextprime
def agen(): # generator of terms
    p = 5
    while True:
        X = (p-1)//2
        Xf = factorial(X)**2
        if all(pow(factorial(Y), 2, p)+1 != p for Y in range(X-1, 0, -1)):
            yield p
        p = nextprime(p)
        while p%4 != 1:
            p = nextprime(p)
print(list(islice(agen(), 5))) # Michael S. Branicky, May 30 2022

Extensions

a(20) and beyond from Michael S. Branicky, May 30 2022