Assune p is an odd prime such that p = 1mod4 (that is p 4k + 1) Prove: if a is a primitive root for p, then so is -a (so, we're assuming ord,(a) = p- 1 and we have to show ord,(-a) p-1, by checking the two conditions of order)

Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

A:

Q: If 2n +1 is prime for some n ≥ 1, prove that n is a power of 2.(Primes of the form 2n +1 are called…

A: Given: 2n+1 is prime for some n≥1. To Prove: n is a power of 2.

Q: Find a formula for 1/2+1/4+1/8+.+ 1/2h by examining the values of this expression for small values…

A:

Q: (a) Show that -163 37 Actually, the following more general result is true: -163 р -163 31 = -1. = -1…

A:

Q: If p > 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p > 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

Q: Use induction to show that n4 + 15 is a multiple of 16 for every positive odd integer n.

A:

Q: Let p be a prime, a ∈ Z. Prove that ap − a is divisible by p.

A:

Q: In the process of proving P(n) by induction, we need to assume that P(n) is true and show that…

A:

Q: Given n > 1, let p be a prime number such that n < p < 2n (by the way, 2n there is always a prime…

A: here we need to expand  by using  nr=n!r!×n-r! and then show that expression obtained have a factor…

Q: Using Fermat’s little theorem to show that 42 | n7 −n, where n is a positive integer.

A:

Q: a)Consider the following data: 3 is a quadratic residue for 23 and 37. 3 is a quadratic nonresidue…

A:

Q: For the following proof, determine which of the statements given below is being proved. Proof.…

A: Contrapositive: The contrapositive of the statement, " If P, then Q" is " If not Q, then not P".

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Prove A prime is irreducible but not conversely.

A: suppose that a is a prime in an integral domain and a=bc. we must show that b or c is a unit. by the…

Q: (c) Does the polynomial x² - 27 = 0 (mod 1271) have any roots? Note that 1271 = 31 - * 41 is not a…

A: We use the Chinese Remainder Theorem to find the solutions to the congruence x2-27≡0(mod1271). Now…

Q: We'll now try to prove the following result in 2 steps: Theorem: Suppose that p is an odd prime. Let…

A:

Q: if p. (144) =(-¹) ove tha pand gare distin (p-1)(g-1) 4 aci

A:

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Show that for every positive

A:

Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: For the pair of integers 315 and 825: (a) Find the prime factorization of each number. (b) Find the…

A: As per our company guidelines we are supposed to answer only first three sub-parts. Kindly repost…

Q: ak written in the form n = pi p²p with a 20 and p, are distinct prime. The factoriz if we order the…

A:

Q: SUI 5. Using mathematical induction, prove the following statement: 3|(22n – 1) for all n > 1

A:

Q: Suppose |G| = p2q2 , where p < q are primes. Prove that if |G| ̸= 36, then nq = 1.

A:

Q: Suppose p and q are odd primes such that p = q + 8a for some positive integer a. () = () = ( ) .…

A: Legendre symbol:  For any prime number p and any integer a that is not divisible by p, the Legendre…

Q: The Fundamental Theorem of Arithmetic states that if n 1 is an integer then e1e2 n = = P₁¹p²/² ek…

A:

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: c.) There is a conjecture called "Goldbach's weak conjecture" (yes, really!) that every odd number…

A: Given: Odd numbers greater than 5. To prove: Every odd number greater than 5 can be written sum of…

Q: For integer values of a, use a proof by contraposition (also called contrapositive or indirect…

A:

Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

A:

Q: Decide which of the following are valid proofs of the following statement: Let n be any integer.…

A: Let n be any integer. Then n3-n is divisible by 3.

Q: p, q, pq | (apq − ap − aq  + a) for all a.

A:

Q: 7 .Assune p is an odd prime such that p ≡ 1mod4 (that is p = 4k+1) Prove: if a is a primitive root…

A: Given: p≡1mod4       ∵p=4k+1

Q: Prove that for every ponitive integer a if aº – 6° is a prime for some positive integer 6 5 = 2 or 5…

A:

Q: (b) Find a prime p> 5 such that 2 and 3 are both primitive roots modulo P.

A: We test all the primes p&gt;5 and see which prime works.

Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: Use Fermat's theorem to show that for any two distinct odd primes P.9, and any integer C: CM-c-c +c…

A: