How can I find all $(a,b,c) \in \mathbb{Z}^3$ such that $a^2+b^2+ab$, $a^2+c^2+ac$ and $b^2+c^2+bc$ are squares ? Thanks !

This question already has an answer here: Why does the natural ring homomorphism induce a surjective group homomorphism of units? 2 answers

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the sequence $A_7$ is $(1,4,5,2,3,6)$. Suppose you truncate the sequence upto the $\alpha p$th term (where $\alpha$ is a very small constant […]

Find all $a,b,c,d\in \mathbb{Z}^+$, which $a^2+b+c+d,b^2+a+c+d,c^2+a+b+d,d^2+a+b+c$ are all perfect squares. I found $(1,1,1,1)$, but I can’t find more. Is $a=b=c=d$ true?

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. I have been searching around math stackexchange […]

(Note: This question has been cross-posted to MO.) Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. For example, $\sigma(6)=1+2+3+6=12$ and $\sigma(28)=1+2+4+7+14+28=56$. Denote the abundancy index $I$ of $x$ by $$I(x)=\frac{\sigma(x)}{x}.$$ If a positive integer $y$ is one of at least two solutions of $$I(y)=\frac{a}{b}$$ for a given rational number $a/b$, […]

I’m in number theory and I currently have these problems assigned as homework. I’ve looked through the sections containing these problems and I’ve solved/proved most of the other problems, but I can’t figure these ones out. For $n>1$, show that every prime divisor of $n!+1$ is an odd integer that is greater than $n$. Assuming […]

e.g., $n = 3$. Clearly, the powers of $2$ modulo $3$ alternate $2, 1$. The powers of $4$ modulo $3$ are all $1$s. So a power tower of $2$s modulo $3$ must get stuck on $1$. Of course this method is inefficient, as $n = 5$ already shows. Is there a formula, or at least […]

My question is the exact same question as the one in this post but I commented on it but it’s from a year ago so I just wanted to bump it and see if I could get a response: Prove that there's no fractions that can't be written in lowest term with Well Ordering Principle […]

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the right Ore condition when for $x,y\in R$ the right principal ideals generated by them have a non-empty intersection. I can infer from what I’ve seen in various […]

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