A Wilson prime is a prime p such that (p−1)!≡−1 mod p^2. Write a procedure which determines all Wilson primes less than10^4

Question

I try to use the for loop to solve this question, but it does not work. And here is what I did.

For[p = 2,p<=10000,(p-1)! = -1 mod (p^2),Print[p]]

I am not sure how to describe "(p−1)!≡−1 mod p^2" in Mathematica. Could you give me some suggestions? Thank you.

Equal is == in infix form, not =. – Daniel Lichtblau Oct 18 '20 at 15:02 — Daniel Lichtblau, Oct 18 '20 at 15:02

Roman · Accepted Answer · 2020-10-18T12:03:12.707

It's probably faster to do a divisibility test than actually calculate the modulus:

Do[If[PrimeQ[p] && Divisible[(p - 1)! + 1, p^2], Sow[p]], {p, 10^4}] // Reap // Last
(*    {{5, 13, 563}}    *)

For is poor style in Mathematica. Use Do or Table instead.

Print is only useful if you want to read the results from the screen. If you want to use the results in further calculations, a Sow/Reap combination is preferrable: Collecting Expressions during Evaluation.

cvgmt · Answer 2 · 2020-10-18T11:23:54.073

5

Select[Range[10000], Mod[(# - 1)! + 1, #^2] == 0 &]

But I think we should test only the prime number,so we modified the above code.

Select[Prime[Range[10000]], Mod[(# - 1)! + 1, #^2] == 0 &]
(* {5, 13, 563} *)

edited Oct 18 '20 at 11:23

answered Oct 18 '20 at 10:54

cvgmt

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Select[Range[10^5], PrimeQ[#] && Mod[(# - 1)! + 1, #^2] == 0 &] // AbsoluteTiming – cvgmt Oct 18 '20 at 13:04

A Wilson prime is a prime p such that (p−1)!≡−1 mod p^2. Write a procedure which determines all Wilson primes less than10^4

2 Answers2