show that no integer of the form n^3+1 is a prime other than 2

Q: how that in the domain of integers of the form a+bsqrt(-17) the factorization…

A: The  objective is to show that 13 is irreducible. The norm of 13 is 13=132=169 So if 13 is reducible…

Q: Prove 10^47+67 is not a prime(use Fermat's). Thank you :)

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Q: True or false: If n is a positive integer with d(n) = 6, then the only possible prime power…

A: n is a positive integer with dn=6.

Q: If n is a positive integer, explain why each prime in the factorization 'n² must occur an even…

A: (a) Let p be a prime number that occurs in the prime factorization of n2.   This implies that p…

Q: Prove that every positive integer n>1 has a prime factorization.

A: Prime factorization is a way of expressing a number as a product of its prime factors.

Q: Is it true that (3 x EN) (x²-3x +20 is a prime)? O True False

A: Is it true that ( there exists x in N)(x^2-3x+20  is a prime ) ?   True False

Q: Compute the prime factorization of:                     gcd(75, 90)                     75 = 3 · 52…

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

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Q: Find all integers n such that (n^2) - 1 is prime.

A: n² - 1 = (n-1)(n+1)  Now, if (n² - 1) needs to be a prime, then, either n-1 is equal to 1 or, n+1 is…

Q: primes

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Q: Which positive integers less than 20 are relatively prime to 20?

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Q: 6. Prove that n, n+2, n +4 are all primes if and only if n = 3.

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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.

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Q: factorization

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Q: Use mathematical induction to prove that 3"7" - 2 is divisible by 8 for every non-negative integer…

A: Initial case:

Q: How Many Positive Integers ≤630 are relatively prime to 630?

A: The given number is .We have to find the number of positive integers are relatively prime to .

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

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: a formal prove that there are infinitely prime numbers.

A: We shall prove it by contradiction. Suppose there are only finitely many primes, say 2, 3,5,··· ,p.…

Q: Jse generating functions to show that every positive integer can be written in exactly one way as a…

A: A generating function is a way to encode an infinite sequence of numbers by treating them as the…

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

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

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Q: Use induction to prove that for all even natural numbers nn, the integer n^3+2n^2−32 is divisible by…

A: Given that , Pn=n3+2n2-32 Prove that Integer Pn is divisible by 8 for all even natural numbers n…

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

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Q: O Determine whether M₁1 = 2¹¹ – 1 is prime by checking primes less than √M₁.

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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: 5. Find the canonical prime factorization and solve for No. of positive divisor and sum of positive…

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Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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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: Compute the prime factorization of:         gcd(699840, 1215000)            699840 = 26 · 37 · 5…

A: We have to find the prime factorization of gcd of two given numbers.

Q: 13. Pier Fermat (1601 – 1665) conjectured that any number F, of the form 2²" + 1 where n is…

A: Fermat number is Fn=22n+1

Q: Find all primes that divide 50!

A: Solution:-

Q: Show that the prime factorization of positive integers is unique

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Q: Find the prime factorization of (2) and of 20!. 10

A: To find the prime factorization of 2010 and of 20!. Prime factorization means we have to represent…

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: Use the PMI to prove that 6" – 1 is divisible by 5 for every integer n > 0.

A: To prove that 6^n-1 is divisible by 5 for every integer greater that or equals to 0.