Factors of a Mersenne number
A Mersenne number is a number in the form of 2P-1. In the search for Mersenne Prime numbers it is advantageous to eliminate exponents by finding a small factor before starting a, potentially lengthy, Lucas-Lehmer test. There are very efficient algorithms for determining if a number divides 2P-1. For example, let's see if 47 divides 223-1. Convert the exponent 23 to binary, you get 10111. Starting with 1, repeatedly square, remove the top bit of the exponent and if 1 multiply squared value by 2, then compute the remainder upon division by 47.
You are encouraged to solve this task according to the task description, using any language you may know.
Remove Optional Square top bit multiply by 2 mod 47 ------------ ------- ------------- ------ 1*1 = 1 1 0111 1*2 = 2 2 2*2 = 4 0 111 no 4 4*4 = 16 1 11 16*2 = 32 32 32*32 = 1024 1 1 1024*2 = 2048 27 27*27 = 729 1 729*2 = 1458 1
Thus, 223 = 1 mod 47. Subtract 1 from both sides. 223-1 = 0 mod 47. Since we've shown that 47 is a factor, 223-1 is not prime. Further properties of Mersenne numbers allow us to refine the process even more. Any factor q of 2P-1 must be of the form 2kp+1. Furthermore, q must be 1 or 7 mod 8. Finally any potential factor q must be prime.
Task: Using the above method find a factor of 2929-1 (aka M929)
Fortran
PROGRAM EXAMPLE IMPLICIT NONE INTEGER :: exponent, factor WRITE(*,*) "Enter exponent of Mersenne number" READ(*,*) exponent factor = Mfactor(exponent) IF (factor == 0) THEN WRITE(*,*) "No Factor found" ELSE WRITE(*,"(A,I0,A,I0)") "M", exponent, " has a factor: ", factor END IF CONTAINS FUNCTION isPrime(number) ! code omitted - see Primality by Trial Division END FUNCTION FUNCTION Mfactor(p) INTEGER :: Mfactor INTEGER, INTENT(IN) :: p INTEGER :: i, k, maxk, msb, n, q DO i = 30, 0 , -1 IF(BTEST(p, i)) THEN msb = i EXIT END IF END DO maxk = 16384 / p ! limit for k to prevent overflow of 32 bit signed integer DO k = 1, maxk q = 2*p*k + 1 IF (.NOT. isPrime(q)) CYCLE IF (MOD(q, 8) /= 1 .AND. MOD(q, 8) /= 7) CYCLE n = 1 DO i = msb, 0, -1 IF (BTEST(p, i)) THEN n = MOD(n*n*2, q) ELSE n = MOD(n*n, q) ENDIF END DO IF (n == 1) THEN Mfactor = q RETURN END IF END DO Mfactor = 0 END FUNCTION END PROGRAM EXAMPLE
Output
M929 has a factor: 13007