In this case n = 618240007109027021
Therefore from the above calculations, we obtain the following;
p = 250387201
Therefore the small prime factors of p-1 are
(2)8Â (3)5Â (5)2Â (23) (7)
2)Â We first select a bound for the factorial.
b = ln (n)
is the acceptable limit.
in this case 43.
Exponent is 43! = 6.0415263063373835637355132068514e+52 (factorial of 43)
then we choose some number to raise to that exponential value.
A number â€˜aâ€™ from 2 to n-1 can be used.
In this case, I chose 2
Then thatâ€™s a factor of n
If not, raise the bound limit.
P = 250387201
//The function to find the prime factors of n is;
public static BigInteger primefactor(BigInteger n) // n is the integer to factorize
public static BigInteger primefactor(BigInteger n) // n is the integer to factorize
static void execA()
static void factor(BigInteger n)
public static void main(String args) //Main method
To factor in a larger value of n, you insert the following method in the code;
execB(); // then call execB() function in main
Factoring n = 642401 using the information;
5161072Â â‰¡ 7 (mod n)
1877222Â â‰¡ 22Â âˆ™ 7 (mod n)
n = 642401
Therefore factor1 := igcd (516107*187722-2*7, n); // n = 642401
factor1 := 1129
and we obtain factor2 by dividing n by factor1
factor2 := n/factor1;
factor2 := 569
Thus 642401= 1129 * 569 which are indeed primes
given that 2(nâˆ’1)Â â‰¡ k2Â _â‰¡Â 1 (modÂ n)
Then we can deduce from â€œFermatâ€™s Little theoremâ€ that n is not indeed a prime
Because 2(nâˆ’1)/2Â _â‰¡ Â±1 (modÂ n) and 2(nâˆ’1)Â â‰¡Â 1 (modÂ n)
then From the â€œbasic factoring principleâ€;Â k2Â â‰¡Â 12Â (modÂ n)
and k _â‰¡ Â±1 (modÂ n)
Therefore we can find a nontrivial factor of n using the formula gcd(k-1, n) or gcd(k+1, n)
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