Euler Prime

Author

Eric W. Weisstein
November 26, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/EulerPrime.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EulerPrime.html.

©2014 Wolfram Research, Inc. except for portions noted otherwise

Enumerate

Position[Table[n^2+n+41,{n,100}],_?PrimeQ]//Flatten//Timing

Select[Table[n^2+n+41,{n,100}],PrimeQ]//Timing

Records

Value

big=;

Compute

http://primes.utm.edu/primes/page.php?id=111195

Prime of the form N = x^2 + x + 41 = 1 mod 163^90000. The Pari-GP code for N is as follows: {D=163; n=90000; M=D^n; B=truncate(sqrt(4-D+O(D^n))); X=50942*M-(B+1)/2; N=X^2+X+41;}

𝒟=163;
n=90000;
M=𝒟^n;
B=Series[Sqrt[4-d],{d,0,n}]

Mod[big,163^90000]

Reduce[x^2+x+41==a&&x>0,x,Reals]

N[big]

IntegerLength[big]

IntegerQ[ind=(1/2)((Sqrt[4 big-163])-1)]

N[ind]