How to show that there are infinitely many prime p of the form p= 1+5k or...

How to show that there are infinitely many prime p of the form p= 1+5k or p=4+5k

Expert Solution

Colby Messinger answered 1 year ago

Prove: There are infinitely many primes of the form 6n − 1 (n is an integer).

Let gcd(a, p) = 1 with p a prime. Show that if a has at least...

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

Provide an example: 1) A sequence with infinitely many terms equal to 1 and infinitely many...

Provide an example: 1) A sequence with infinitely many terms equal to 1 and infinitely many terms that are not equal to 1 that is convergent. 2) A sequence that converges to 1 and has exactly one term equal to 1. 3) A sequence that converges to 1, but all of its terms are irrational numbers.

Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that...

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that (p − 1)2 = 1 (mod p) 2. Is the above true for any number (not necessarily prime)? 3. Show that the equation a 2 − 1 = 0, has only two solutions mod p. 4. Consider (p − 1)!. Show that (p − 1)! = −1 (mod p) Remark: Think about what are the values of inverses of 1, 2, . . ....

Show that if G is a group of order np where p is prime and 1...

Show that if G is a group of order np where p is prime and 1 < n < p, then G is not simple. (Please do not use Sylow theorem)

Discrete math problem: Prove that there are infinitely many primes of form 4n+3.

Show by induction that if a prime p divides a product of n numbers, then it...

Show by induction that if a prime p divides a product of n numbers, then it divides at least one of the numbers. Number theory course. Please, I want a clear and neat and readable answer.

Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces...

Let p be an odd prime (i.e., any prime other than 2). Form two vector spaces V1, V2 over Fp (prime field of order p) with bases corresponding to the edges and faces of an icosahedron (so that V1 has dimension 30 and V2 has dimension 20). Let T : V1 → V2 be the linear transformation defined as follows: given a vector v ∈ V1, T(v) is the vector in V2 whose component corresponding to a given face is...

24. Show that (x ^p) − x has p distinct zeros in Zp, for any prime...

24. Show that (x ^p) − x has p distinct zeros in Zp, for any prime p. Conclude that (x ^p) − x = x(x − 1)(x − 2)· · ·(x − (p − 1)). (this is not as simple as showing that each element in Zp is a root -- after all, we've seen that in Z6[x], the polynomial x^2-5x has 4 roots, 0, 5, 2, and 3, but x^2-5 is not equal to (x-0)(x-5)(x-2)(x-3))

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