Articles of elementary number theory

If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.

Problem is as stated in the title. Source is Larson’s ‘Problem Solving through Problems’. I’ve tried all kinds of factorizations with this trying to get it to the form $$k^2l^2$$ but nothing’s clicking. I tried Bézout but the same expression can be written as $$a^2 + (a^2 + 1)(a+1)^2$$ which would imply that there is […]

Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime

How can I show that $(n-1)!$ is congruent to $-1 \pmod{n}$ if and only if $n$ is prime? Thanks.

suppose $\gcd(a,b)= 1$ and $a$ divides $bc$. Show that $a$ must divide $c$.

Well I thought this is obvious. since $\gcd(a,b)=1$, then we have that $a$ does not divide $b$ AND $a$ divides $bc$. this implies that $a$ divides $c$. done. but apparently this is wrong. help explain why this way is wrong please. the question tells you give me two relatively prime numbers $a$ and $b$ such […]

Integer solutions for $x^2-y^3 = 23$.

As the title stated, I am wondering the integers $x,y$ that satisfy the equation $x^2-y^3 = 23$.

Number of roots of polynomials in $\mathbb Z/p \mathbb Z $

I was given this proposition but I was never able to prove it. Does anyone know how to solve this? Let $f$ be a polynomial in $\mathbb{Z}/p\mathbb{Z}[x]$, where $p$ is prime. Then $f$ has at most $\deg f$ roots.

How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$?

Show that if $a$ and $b$ are positive integers, then $$\left(a +\frac12\right)^n + \left(b+\frac{1}{2}\right)^n$$is an integer for only finitely many positive integers $n$. I tried hard but nothing seems to work. 🙁

Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions

Prove that $x^{2} \equiv 1 \pmod{2^k}$ has exactly four incongruent solutions. My attempt: We have, $x^2 – 1 = (x – 1) \times (x + 1)$, then $(x – 1)(x + 1) \equiv 0 \pmod{2^k}$ which implies, $2^k|(x – 1)$ or $2^k|(x + 1) \implies x \equiv \pm 1 \pmod{2^k} (1)$ Furthermore, $2^{k-1} \equiv 0 […]

Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$

Let $a,b,c \in \mathbb N$ find integer in the form: $$I=\frac{a}{b+c} + \frac{b}{c+a} + \frac{c} {a+b}$$ Using Nesbitt’s inequality: $I \ge \frac 32$ I am trying to prove $I \le 2$ to implies there $\nexists \ a,b,c$ such that $I\in \mathbb Z$: $$I\le 2 \\ \iff c{a}^{2}+3\,acb+{a}^{3}+{a}^{2}b+a{b}^{2}+{b}^{3}+c{b}^{2}+{c}^{2}b+{c }^{3}+{c}^{2}a-2\, \left( b+c \right) \left( c+a \right) \left( […]

prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$

I’m trying to prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$ I showed that both $n,m$ divides $nm/\gcd(n,m)$ but I can’t prove that it is the smallest number. Any help will be appreciated.

Show that 13 divides $2^{70}+3^{70}$

Show that $13$ divides $2^{70} + 3^{70}$. My main problem here is trying to figure out how to separate the two elements in the sum, and then use Fermat’s little theorem. So how can I separate the two? Thanks!