For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

Q: Prove by simple induction on n that 2^n > n

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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: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

Q: Let р be a prime such that p = 22" + 1, for some n € N, with n > 1. (a) Determine if 5 is a…

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Q: 9. Show that there are infinitely many primes of the form 16k + 1. More generally, show that for any…

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Q: Determine the smallest integer k > 1 such that ak = 1 modulo 2465 for all in- tegers a which are…

A: A composite number m is called Carmichael number if for all integers b coprime with m, bm-1≡1modm…

Q: Prove that there are infinitely many primes.

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Q: Prove, that if 2m + 1 is a prime number, then m must be a power of 2 (m = 2n, for some n ∈ N). Prime…

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Q: 8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1…

A: a) Given that p and q are odd primes and qap-1. So, ap≡1mod q If p=1 then  a≡1mod q So, qa-1 If p≠1…

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: Use induction to prove that 2*3^(n-1) is less than or equal to 2n choose n (picture attached)

A: We have to prove 2⋅3n−1≤2nn by using of principal of mathematical induction, where n≥0 Recall that…

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

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Q: Suppose g is a primitive root mod p (with p an odd prime) such that g +1 mod p? Prove g must be a…

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

Q: gcd (k, Pal)=- x* =b (mod P). Theorm: If p aned q are dis tinbed primes non of which divides b, and…

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Q: Let p be an odd prime, and let g be a primitive root modulo p. Prove that -g is a primitive root…

A: Given : p= odd prime g= primitive root modulo p

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: Suppose that residue(modp) 8) Let odd prime P. atzis Show that has be an not Diophantine…

A: Given that a is not a quadratic residue mod p. That is there does not exists an integer z such that,…

Q: 8. Use Euler's Theorem to find a number a between 0 and 9 such that a is congruent to 7 1000 modulo…

A: We know according to  Euler's Theorem  that  xϕ(n)≡1(mod n) where the function ϕ(n) counts the…

Q: Let p be an odd prime number and an integer g a primitive root with respect to mod p. So is the…

A: This is not a true statement. Since If p be an odd prime and g be a primitive root modulo p. Then…

Q: Prove the formula () - (*: ) - (**) ---() -(*) r + n + 1 n +• r +1 by induction on n (for r…

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Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

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Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

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Q: Euler's Theorem states that if n is a positive integer and a is relatively prime to n, then ar(n) =…

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Q: Find the modular residue of the product. 9,512,667 2,183,321 (mod 5) 9,512,667 2,183,321 (mod 5) =…

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Q: Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Given that p is prime . Let us take p=19 By Fermat's little theorem, " If p is a prime and a is any…

Q: Use Dirichlet's theorem to show that there are infinitely many primes whose dicimal expansion ends…

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Q: For any n e N, 3 divides n – n. -

A: Prove using Mathematical of induction

Q: If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

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Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: Find (with proof) all positive primes p such that 217p is a Carmichael number.

A: Carmichael number - A number n is a Carmichael number if and only if n=p1p2…pj , a product of at…

Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

Q: more general conjecture is that there exists infinitely many primes of the form n^2 + 1; for example…

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Q: Let p be prime. Show that C) mod p = C for 0 <k< p What does this imply about the binomial…

A: See the following below.

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Prove by induction that the sum of the first n odd numbers is n². In other words, prove E1(2i – 1) =…

A: To prove by induction that the sum of the first natural numbers is n^2.

Q: 5. Use ordinary induction to prove that for every positive integer n, n°-n is a multiple of 6. Only…

A: To prove that  For every positive integer n, n^3-n is a multiple of 6 by using induction method.

Q: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

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Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…