Question

*If g(x) = x + x^2, prove that g(n) − 4g(m) = 0 has
no solutions for positive integers m and n.*

Answer #1

Prove by induction that
2 x 1! + 5 x 2! + 10 x 3! +...+ (n2 + 1) n! = n (n + 1)! For all
positive integers n

Prove the following statements:
1- If m and n are relatively prime,
then for any x belongs, Z there are integers a; b such that
x = am + bn
2- For every n belongs N, the number (n^3 + 2) is not divisible
by 4.

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Prove that if x ∈ Zn − {0} and x has no common divisor with n
greater than 1, then x has a multiplicative inverse in (Zn − {0},
·n).
State the theorem about Euler’s φ function and show why this
fact implies it.

Using induction prove that for all positive integers n, n^2−n is
even.

Prove that for all positive integers n,
(1^3) + (2^3) + ... + (n^3) = (1+2+...+n)^2

Prove: If f(x) = anx^n + an−1x^n−1 + ··· + a1x + a0 has integer
coefficients with an ? 0 ? a0 and there are relatively prime
integers p, q ∈ Z with f ? p ? = 0, then p | a0 and q | an . [Hint:
Clear denominators.]

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

Prove that the complete set of solutions to the congruence x
4 − x 2 + x + 3 ≡ 0 (mod 51) is given by {x ∈
Z: x ≡ 36 (mod 51)}

Prove that there are no solutions in positive integers to the
equation x3 + y3 = 100

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