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

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

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

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

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

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Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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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: 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…

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Q: Using direct proof method, prove the following. If n is an even integer then n²+ 13 is odd.

A: n is even, so take n=2k and put on the expression.

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: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: Show that yn for all n=1, 2₁- ✓

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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: using

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

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

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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: 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.…

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

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Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: 2n 3. Let a be an odd integer. Prove by induction that a²" - 1 is divisible by 2n+1 for every…

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Q: Let c = 2^100 and p be the smallest prime such that p >= c. How many composite numbers are between c…

A: Let c=2100 And p=min p is prime:p≥c,p∈ℤ Claim: Total number of composite number between c & p,…

Q: If 3x = 6 mod 9, when -2 <x< 8. Then O x=(-1, 3, 5} x=(-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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Q: Show that Consider the Fibonacci numbers {F} defined by F₁ = 1, F₂ = 1, and Fn+2= Fn+1 + Fn. - Fn =…

A: Introduction: In mathematics, a recurrence relation is a significant connection that expresses the…

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: 2. If p > q > 5 and P and q are both primes, prove that 24 p - q² |

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Q: Prove that 3n^2 + 1is divisible by 4 whenever n is an odd number.

A: Using induction method we have to prove that 3n^2 +1is divisible by 4 where n is odd number.

Q: Let m = p^3 qr^3 and n = p^2 r^5, where p,q,r are distinct primes. Find each of the following. a.…

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

A: Given statement is Here we can select 38 even positive integers all are less than 1000. let…

Q: If 3x = 6 mod 9, when -2 < x< 8. Then O x={-1, 3, 5} x-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

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Q: In the question that follows, the "Units Digit" or the "Unity Digit"of a number is the last digit in…

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Q: Prove that 3" +4" ≤5" for all integers n ≥ 2.

A: The given problem is to prove the following inequality 3n+4n&lt; 5n for all the integers n&gt;2.