Articles of number theory

If $n_j = p_1\cdot \ldots \cdot p_t – \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$

Show that if $p_1,\ldots p_t$ are the first $t$ prime numbers, and $n_j = p_1\cdot \ldots \cdot p_t – \frac{p_1\cdot \ldots \cdot p_t}{p_j}$, then $\phi(n_j)=\phi(n_k)$ for $1 \leq j,k \leq t$ and conclude that the equation $\phi(x)=m$ has infinitely many solutions. Here $\phi(\cdot)$ is the Euler Totient function. I am really stuck on this one. […]

Statements with rare counter-examples

This question already has an answer here: Examples of patterns that eventually fail 34 answers

If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$

Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$. Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$. (problem composed by Laurentiu Panaitopol) So far no idea.

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I believe many of you know some nice proofs of this, can you please share it with us?

Generate solutions of Quadratic Diophantine Equation

Recently I’ve asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: The quadratic diophantine $ k^2 – 1 = 5(m^2 – 1)$ Here’s the answer: $$ x_n + y_n \sqrt{5} = \left(x_{n-1} + y_{n-1} \sqrt{5}\right)\left(\frac {3 + \sqrt{5}}{2}\right)^n$$ Actually it worked for my equation, which […]

Why is $e^{\pi \sqrt{163}}$ almost an integer?

The fact that Ramanujan’s Constant $e^{\pi \sqrt{163}}$ is almost an integer ($262 537 412 640 768 743.99999999999925…$) doesn’t seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it’s almost-but-not-quite an integer in layman’s terms (I’m not a mathematician)?

Is '10' a magical number or I am missing something?

It’s a hilarious witty joke that points out how every base is ’10’ in its base. Like, 2 = 10 (base 2) 8 = 10 (base 8) My question is if whoever invented the decimal system had chosen 9 numbers or 11, or whatever, would this still be applicable? I am confused – Is 10 […]

Linear independence of roots over Q

Let $p_1,\ldots,p_k$ be $k$ distinct primes (in $\mathbb{N}$) and $n>1$. Is it true that $[\mathbb{Q}(\sqrt[n]{p_1},\ldots,\sqrt[n]{p_k}):\mathbb{Q}]=n^k$? (all the roots are in $\mathbb{R}^+$) Iurie Boreico proved here that a linear combination $\sum q_i\sqrt[n]{a_i}$ with positive rational coefficients $q_i$ (and no $\sqrt[n]{a_i}\in\mathbb{Q}$) can’t be rational, but this question seems to be more difficult..

Legendre symbol, second supplementary law

$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$$ how did they get the exponent. May be from Gauss lemma, but how. Suppose we have a = 2 and p = 11. Then n = 3 (6,8,10), but not $$15 = (11^2-1)/8$$ n is a way to compute Legendre symbols from Gauss lemma: $$\left(\frac{a}{p}\right) = (-1)^n$$

Can a prime in a Dedekind domain be contained in the union of the other prime ideals?

Suppose $R$ is a Dedekind domain with a infinite number of prime ideals. Let $P$ be one of the nonzero prime ideals, and let $U$ be the union of all the other prime ideals except $P$. Is it possible for $P\subset U$? As a remark, if there were only finitely many prime ideals in $R$, […]