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15 Questions

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7	Find all solutions of the equation $x! + y! = z!$ [duplicate]
5	If $\phi(n)$ divides $n-1$, prove that $n$ is a product of distinct prime numbers
3	Proof showing there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
3	Find all values of $n$ greater than or equal to 1 for which $n! + (n + 1)! + (n + 2)!$ is equal to a perfect square.
3	Prove that either $m$ divides $n$ or $n$ divides $m$ given that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$?

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+8	Do there exist infinitely many positive integers n such that ϕ(n) = n/4?
+3	Find n if $ϕ(n) = 2^{6} \times 17$ and $τ(n) = 12$.
+10	Proof showing there exists a sequence of $m$ consecutive natural numbers which contains exactly $n$ primes.
-2	If all primes in the prime factor decomposistion of both $m , n$ occur at powers $\geq 2$, and $ \phi (m) = \phi (n)$, then m = n.

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0 number-theory × 15	0 divisibility
0 factorial × 2	0 elementary-number-theory
0 natural-numbers	0 prime-numbers

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