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In point of fact, it is well-known that if $p$ is a prime number and $a, n$ are integers that are not divisible by $p$, then $p \mid \Phi_{n}(a)$ implies $p\equiv 1 \pmod{n}$.

The notion of natural density gives a suficient condition. Namely, one can prove that if $A \subseteq \mathbb{N}$ has positive upper natural density then $\sum_{a \in A} \frac{1}{a}$ diverges. This co …

There are only three terms of the Fibonacci sequence which are equal to a factorial: $$F_{1}= F_{2} = 0!=1! \quad \mbox{and} \quad F_{3}=2!$$ What is more, F. Luca proved in this paper (published in …

Short answer: yes. In „Zur Verallgemeinerung des Bertrandschen Postulates, daß zwischen $x$ und $2x$ stets Primzahlen liegen" (Mathematische Zeitschrift, 34 (1932), pp. 505-526), R. Breusch proved tha …

Let us suppose that $x^{n}+y^{n}= z^{n}$ with $x, y,$ and $z$ relatively prime. By the abc conjecture, $|x^{n}|\ll |xyz|^{1+\epsilon}$, $|y^{n}|\ll |xyz|^{1+\epsilon}$, and $|z^{n}|\ll |xyz|^{1+\epsil …

It is well-known that A: The series of the reciprocals of the primes diverges My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers. Property A …

Wow! I have been thinking about this problem, too. I can prove that for every odd natural number $d$ there exists infinitely many $n \in \mathbb{N}$ such that $n^{d}+1 \mid n!$. Here you have my gorge …

As I was browsing through the problems & solutions department of the American Mathematical Monthly the other day, I found out that, in his solution to problem E-2332 [1972, 87], which was published on …

It has to be noted that we can also give a (conditional) proof of Bertrand's Postulate assuming the veracity of Goldbach's Conjecture. This was the subject matter of a filler piece that appeared on pa …

This one will be quick... Wonder if anybody knows or remembers the title of the paper in which Karatsuba introduced his approach at Burgess's bound on character sums. Thanks for your support. EDIT. …

I am to elaborate a bit on how it is that the answer I left in this previous question implies a positive answer to Mr. Recamán's question. By elementary means (i.e., without resorting to any complex …

W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in integers $x>1, y>1, n>2$, except when $n=4, x=7$ and $n=5, x=3$. Since your equation …

Erdős and Diamond proved in [1] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the f …

P. Erdős and Leon Alaoglu proved in [1] that for every $\epsilon > 0$ the inequality $\phi(\sigma(n)) < \epsilon \cdot n$ holds for every $n \in \mathbb{N}$, except for a set of density $0$. C. L. me …

Hello, Is anyone here aware of a well-motivated exposition of the Bourgain-Glibichuk-Konyagin estimate for exponential sums (or Gauss sums) over multiplicative subgroups? If any of you has a write-up …