3) Support the p is a that Show least one Prove then iss prime 27. ท (1^2) = (2+1) = 1 for at number {1, 2, ---,9} مروز n in the set that if p is طر an odd prime x² = 2 (mod p) p=1 or 7 (mod 8). has solutions

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Prove that Va - Vb is irrational number if 'a' and 'b’ are prime number?

A: Given To prove that a-b is irrational number if a and b are prime number.

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

Q: b"=1 (mod n) is valid for which of the following pair (b,n) (9.7,1729) (97,1729) (-97,1729)…

A: Answer

Q: Prove that the quadratic congruence 6x^(2) + 5x + 1≡  0 (mod p) has a solution for every prime p,…

A: We want to prove that the quadratic congruence 6x2+5x+1≡0 mod p has a solution for every prime p…

Q: 7) Let p be an odd prime, find if a) p =1 mod 8

A:

Q: . Numerical evidence makes it plausible that there are infinitely many primes p such that p + 50 is…

A:

Q: True or false: If p is an odd prime, then (p-1)! + 2^(p-1) congruent 0(mod p). Justify your answer.

A: p is an odd prime.

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: L-) Find all in tegers such thet 5x = 12 (mod 7) 10x = 4 (mod 11) E 16 (mod 10d ond 1occ < x < 3000

A: To determine: All the integers x such that 5x≡12 (mod 7)10x≡4(mod11)7x≡16(mod 10)and 1000≤x≤3000

Q: Let p (> 0) be the first of the n arithmetic means betw numbers and q (>0) the first of n harmonic…

A:

Q: Let m be an element of N, be a relatively prime to 5. Show that 5|(m^2 + 1)(m^2 + 4)

A:

Q: Suppose that a and b are integers. a ≡ 7 (mod 13) and b ≡ 9 (mod 13). Find the integer c such that 0…

A: The given problem is to find the modular values of c for the given conditions with given integers a…

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: Show that II (1-) < II p° perfect = n = p*||n p*||n Hint: The following formula should prove useful:…

A:

Q: if p is any prime and a is any integer, then p/(a^p+(p-1)!a) and p/(a+(p-1)!a^p) Prove equivalent…

A: To prove the equivalence of the two statements, we need to show that if the first statement is true,…

Q: III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1,…

A: We shall show that no such positive integers exist.

Q: Suppose that p is a prime number. Given this information prove that if x is an integer such that x^2…

A:

Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

A:

Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

A:

Q: Prove disprove that or characteristics of simple xing is either prime . žero or

A: To prove or disprove that characteristics of a simple ring is either zero or prime

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: determine what number the sum is congruent to in mod 3, 6+4 = __ (mod 3)

A: First we add the numbers, then check for the remainder on dividing the result by 3 and that will be…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A:

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡−1 mod 7

A: We will prove the given statement.

Q: S²₁ Sinmxsinhxbx = 2 x if m=n So min

A: To prove:- ∫-ππsinmxsinnxdx=0if m≠nπif m=n

Q: Let p be a prime. Find the least residue mod p of 1^p + 2^p + ... + (p-1)^p

A: The given problem is to find the least residue mod p of 1p+2p+....+(p-1)p.

Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

A:

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Compute all least residue solutions modulo 12 of 8x-4 congruent to 0(mod 12)

A:

Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: Given below an step by step solution

Q: てheorem' Let p be aprime number, a integer (ap)-1 0nd pacitive integer. In this cose; ) #1 (mod p)…

A: We need to use the above given two theorems to find the solution for the given congruence

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: We have to find few terms of the given sequences.

Q: Find the nonnegative integer a less than 28 represented by each of these pairs, where each pair…

A: According to the given information, it is required to find the non-negative integer a less than 28…

Q: Let S be the set of numbers of the form { ((-1)^(n + 1)) (n - 1)/(n + 1) } where n is an element of…

A:

Q: What sequence of pseudorandom numbers is generated using the linear congruential generator xn+1…

A: Determine the sequence of pseudorandom numbers generated using the linear congruential generator…

Q: 5 how that Cm d n+2 Che prine 4C(n-1)! +)+n = 0 mod @ncnt2)

A: We show that the n and n+2 are primes If and only if  4((n-1)!+1)+n≡0 mod n(n+2) We take n as prime…

Q: prove the following using either direct or indirect proof  a. if a is an odd integer, then…

A:

Q: Prove that 52-1 is divisible by 8 for all NEN 1) show for N=1 5263-1-N 52-1-21 =3 2(1) 24 8 8…

A:

Q: ) How many,8-distinct/digit numbers can be formed if the numbers áre amultiple of/5 and contain the…

A:

Q: Find all solutions of the congruence x^2 = 208 (mod 7368787).

A: As per the question we are given the following quadratic congruence equation : x2 ≡ 208 (mod…