ANSWER :
Mersenne primes are of the form : 2^n - 1 ; n = natural number ≥ 1 .
(excluding 2^4 - 1 = 15 which is not a prime).
Fermat primes are of the form : 2^n + 1 ; n = any natural number.
(excluding 2^3 + 1 = 9 which is not a prime)
Fermat’s more popular form : 2*2^n + 1 ; n = any natural number.
4. (a) [3] Let p be prime and let M, denote the number 2P – 1. The number M, is called a Mersenne number, and if it is prime, it is called a Mersenne prime. There is a test, called the Lucas-Lehmer Test, that gives a necessary and sufficient condition for My to be prime. It is always used to verify that a Mersenne number, suspected of being prime, is indeed a Mersenne prime. Give the statement of this test....
Problem 1 Define Mp 2" - 1 for a prime number p. The number Mp is called a Mersenne number. For example, M2 3, M37, M5 31, and M7127, and they are all prime numbers. When Mp is a prime number, it is called a Mersenne prime (1) Is M11 a prime number? (2) Is M13 a prime number?
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P) 4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
Find a Carmichael number of the form 7.23.p, where p is prime.
negate: (b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is called composite if it is not prime.) (c) For every integer n > 0, there exists a prime number p such that n S p < 2n. (b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.
please post clear picture or solution. Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P, p implies p | r. A positive integer z is called a perfect power if there exist a prime number p and a natural number n such that p". An Achilles number is one that is powerful but is not a perfect power. For example, 72 is an Achilles number. Prove that if a and b...
Question 6 (optional) For positive integers p 2 2, Wilson's Theorem states that p is a prime if and only if (p-1)!-1 (mod p) (a) Prove Wilson's Theorem (b) Discuss whether Wilson's Theorem is suitable as a primality test for finding primes to use with RSA. Question 6 (optional) For positive integers p 2 2, Wilson's Theorem states that p is a prime if and only if (p-1)!-1 (mod p) (a) Prove Wilson's Theorem (b) Discuss whether Wilson's Theorem is...
10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0 10. Let p be a prime number. We know that p divides (p- 1)!+1. Show that if p> 5 then (p- 1)!+1 is never of the form pë where e e Z0
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...