Pocklington's theorem

The Pocklington's theorem invented by H. C. Pocklington in 1914, which is a primality test for the number N, states:

Let [math]\displaystyle{ N-1 = q^k\,R }[/math] where q is a prime which does not divide R. If there is an integer a such that [math]\displaystyle{ a^{n-1}\,\equiv\,1\,\pmod{n} }[/math] and [math]\displaystyle{ gcd(a^{(N-1)/q}-1,N) = 1 }[/math], then each prime factor q of N has the form [math]\displaystyle{ q^kr+1 }[/math].

From the previous theorem it can be deduced a primality test when only a partial factorization of N-1 is known:

Let [math]\displaystyle{ N = fr + 1 }[/math], where [math]\displaystyle{ 0\,\lt \,r\,\le \,f+1 }[/math]. The value N is prime if, for every prime divisor p of f, there is an integer x_p such that:

[math]\displaystyle{ x_p^{N-1}\,\equiv\,1\,\pmod{N} }[/math]

[math]\displaystyle{ gcd(x_p^{(N-1)/p}\,-1,\,N) = 1 }[/math]

If f < r, Pocklington's theorem does not return a primality proof, but there is still some hope of proving primality. The theorem implies that every factor p of N satisfies

[math]\displaystyle{ p \equiv 1 \pmod{f} }[/math]

There are several algorithms which can be used to finish the proof, provided f is big enough. If [math]\displaystyle{ f^3 \gt n }[/math], Selfridge's test can be used to prove primality. If [math]\displaystyle{ f^{10} \gt n^3 }[/math], the Konyagin-Pomerance Test can be used. If [math]\displaystyle{ f^4 \gt n }[/math], there is a polynomial-time algorithm, due to Coppersmith and Howgrave-Graham for carrying out a complete search for divisors congruent to 1 modulo f.

Each of these algorithms is more complicated than the last, so it is best to use the simplest one which will work for the value of f you have. For [math]\displaystyle{ f^4 \lt n }[/math], there is no known algorithm which enhances Pocklington's theorem.

External links

• Pocklington primality test
• Primality proving program based on Pocklington's theorem by Makoto Kamada. It includes a program written in C language that proves primality of numbers of several thousand digits using this theorem.