Let k be a positive integer. Prove that if 2k +1 is prime, then k = 2". The integer F₁ = 22" +1 is called the nth Fermat number. Primes of the form 22" +1 are called Fermat primes. Show that F₁ is prime for n = 1, 2, 3, 4.

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: 6 First Last Prove that 2 1² +2²+...+n² = n(n+i) (²n+1)/6) for positive integer. n. www

A:

Q: Show that for all positive integers m and n, :(). m+n т+n m +n (m+1) . n. m т+1

A:

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: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: 8. Prove that n!+1 and (n+1)!+1 are integer n. relatively prime for every positive

A: Given, and are two numbers and to prove that they are relatively prime.

Q: Let x be an integer. Prove that, if x mod 9 = 5, then 9|(x^2-2x+3)

A: Given: x mod 9=5 then, 9|x2-2x+3|

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: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: If a 2 × 2 matrix A is non-invertible, then the image of the unit square under the linear…

A: False Reason : Here the image of unit square image

Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

A:

Q: Let p be a prime number and a be a positive integer. Prove that p | (a^p - (p-2)!a).

A:

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

A:

Q: Show that if n is an even perfect number, then n = 2P-'Mp for some Mersenne prime Mp 2P – 1.

A:

Q: Prove or disprove the following statement: There are prime numbers a, b, c such that a^2+ b^2= c^2.

A:

Q: primes

A:

Q: Find the least residue of  -5 -46 modulo 14.

A:

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)

A:

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: 7. Prove that n, n +4, n +8 are all primes if and only if n = 3.

A:

Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: are 51 software students taking Let pi, p2,.. Psi be their final cam scores. Hence, cach p, is an…

A: Given that p1, p2, p3,..., p51 are the final exam scores of 51 software students who are taking MATH…

Q: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

A:

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

A:

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A:

Q: We select 38 even positive integers, all less than 1000. Prove that there will be two of them whose…

A: Introduction: Pigeonhole principle: The Pigeonhole Principle is a fundamental principle in…

Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

A:

Q: Find the amount of money owed at the end of 5 years if $60,000 is borrowed at 7% per year compounded…

A: Here we have to find amount of money at the end of 5 years if money borrowed is $60, 000 at 7% rate…

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

A:

Q: Which of the following gives the Binet form of the nth Fibonacci number? O Fn = Fn = Fn = Fn =…

A:

Q: Find all the primes less than 50 with primitive root 3.

A: A primitive root modulo a prime number is an integer such that every non-zero integer less than…

Q: 10 Solve 3Y=2 (mod 7) 8 Salutions exist or show that

A: We have to solve given problem:

Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: TRUE or FALSE: 1. There is no integer n such that 4 | (n^2-2) 2. The GCD of all prime numbers is 1.…

A:

Q: 2) Prove that 4 divides 7" -3" for eachn EN

A:

Q: Let c be an element of N, and p be a prime. Prove that if c is greater than or equal to 3, then c^p…

A:

Q: Denote by an the nth positive integer that is not a perfect square. For example, we have…

A: