Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides...

Show that is [(n^2) + 2] and [(n^2) - 2] are both primes, then 3 divides n

Expert Solution

Colby Messinger answered 2 years ago

Prove that 3 divides n^3 −n for all n ≥ 1.

Show by induction that if a prime p divides a product of n numbers, then it...

Show by induction that if a prime p divides a product of n numbers, then it divides at least one of the numbers. Number theory course. Please, I want a clear and neat and readable answer.

(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this...

(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this is indeed the correct limit. (b) Use an epsilon, N argument to show that {1/(n^(1/2))} converges to 0. (c) Let k be a positive integer. Use an epsilon, N argument to show that {a/(n^(1/k))} converges to 0. (d) Show that if {Xn} converges to x, then the sequence {Xn^3} converges to x^3. This has to be an epsilon, N argument [Hint: Use the difference...

For a given positive integer n, output the first n primes. Example: n=3, output: 2,3,5; n=7,...

For a given positive integer n, output the first n primes. Example: n=3, output: 2,3,5; n=7, output: 2,3,5,7,11,13,17. In Java please

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2...

Prove: There are inﬁnitely many primes congruent to 3 modulo 8. Hint: Consider N = (p1p2···pr)2 + 2.

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n...

prove or disprove .if n is a non negative integer, then 5 divides 2 ⋅ 4^n + 3⋅9^n.

Use a direct proof to prove that 6 divides (n^3)-n whenever n is a non-negative integer.

Show that (a)Sn=<(1 2),(1 3),……(1 n)>. (b)Sn=<(1 2),(2 3),……(n-1 n)> (c)Sn=<(1 2),(1 2 …… n-1 n)>

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n...

Show that (1 + 2 +. . .+n)2 > 12 +. . .+ n2, for n ≥ 2.

For a given integer n > 1, list all primes not exceeding n. Example: n=10, output:...

For a given integer n > 1, list all primes not exceeding n. Example: n=10, output: 2,3,5,7 n=16, output: 2,3,5,7,11,13 In Java please

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