# Questions tagged [square-numbers]

This tag is for questions involving square number. A non-negative integer $n$ is a square number if $n = k^2$ for some integer $k$.

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### It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square

Prove that it is impossible for three consecutive squares to sum to another perfect square. I have tried for the three numbers $x-1$, $x$, and $x+1$.
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### Conjecture: Is $100$ the only square number of the form $a^b+b^a$?

Conjecture $100$ is the only square number of the form $a^b+b^a$ for integers $b>a>1$. In other words, $(a,b)=(2,6)$ is the only solution. Can we prove/disprove this? Observations The ...
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### Korselt numbers; definitions and properties [closed]

If $N$ is an $\alpha$-Korselt number, is $N$ must always be a squarefree , or there is cases can be have a square prime factor.
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### Korselt numbers and sets [closed]

When $N$ is a $K_\alpha$-number($\alpha$-Korselt number), is $N$ must always be a squarefree composite number? or $N$ may be contain a square prime factor?
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### packing uniform cuboids into regular cube

I have a pile of uniform cuboids, of side a,b,c. I would like to make a regular cube. The sides are not in harmonic ratio ( cf. de Bruijn) but are in fact white sugar lumps. The minimum regular cube ...
### If $q$ is prime, can $\sigma(q^{k-1})$ and $\sigma(q^k)/2$ be both squares when $q \equiv 1 \pmod 4$ and $k \equiv 1 \pmod 4$?
This is related to this earlier MSE question. In particular, it appears that there is already a proof for the equivalence $$\sigma(q^{k-1}) \text{ is a square } \iff k = 1.$$ Let $\sigma(x)$ ...