# Questions tagged [square-numbers]

This tag is for questions involving square numbers. A non-negative integer $n$ is called a square number if $n = k^2$ for some integer $k$. Consider using with the [elementary-number-theory] or [number-theory] tags.

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### Find $a$ for which $(a-3)(a-7)$ is a perfect square

The only solutions seem to be 3 and 7 but I can't prove that there are no others. Context: Find every value for integer a, for which $x^2-(a+5)x+5a+1$ expression can be factored as $(x+b)(x+c)$ where ...
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### Exists a perfect square that divided by 6 equals a prime number? [closed]

I recently took a test and a question came up that asked the question: ...
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### Sequence of squares which can't be written as the sum of a smaller non-zero square and twice a triangular number

Are there infinitely many squares which cannot be written as the sum of a smaller non-zero square and twice a triangular number? In other words, is the list given at https://oeis.org/A230312 infinite? ...
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### Writing 2024 as the sum of 3 and 4 squares

I'm currently taking a course in number theory and we've just seen that any number can be written as the sum of the 4 squares, and that numbers can be written as the sum of 3 if they aren't of a ...
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### How to check if something is perfect square or not.

While finding eigenvalues of a particular matrix, I end up with the following: $\sqrt{p^{2\alpha}-4p^{\alpha}+8p^{\alpha-1}+4}$, where $p$ is an odd prime and $\alpha \geq 1$. The next step is to ...
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### Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs

the question Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs. my idea I tried grouping them in a whole perfect square or I tried using formulas such ...
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### Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$

the question Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $$5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$$ my idea I tried grouping them in a ...
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### Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

The question Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$. My idea $a^2+3b^2=25n$ By using modular arithmetic( a perfect square can be only ...
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### Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime.

question Show that the number $2^{2^{2n+1}}+2^{2^{2n}}+1$ is not prime, for any nonzero natural number $n$. my idea $2^{2^{2n+1}}+2^{2^{2n}}+1 = 2^{2^{2n}}*5+1$ i did this by giving common factor. ...
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### Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$?

Except for $1$, are there any perfect powers in the sequence $1,21,321,4321,\cdots$? This is sequence OEIS A000422, the concatenation of positive integers from $n$ down to $1$. If there is any perfect ...
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### Determine the nonzero natural numbers $a$ and $b$ for which the number $n=\sqrt{a^{2}+6b}+\sqrt{b^{2} +6a}$ is rational.

Question Determine the nonzero natural numbers $a$ and $b$ for which the number $n=\sqrt{a^{2}+6b}+\sqrt{b^{2} +6a}$ is rational. My idea First of all because $a$ and $b$ are natural this means that ...
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### If $c\ge b\ge a$ and $c\mid a^2,b^2$, and $a$ is the minimal number satisfy $c\mid a^2$, does $a$ necessarily divide $b$? [duplicate]
Assume $a$, $b$, $c$ are positive numbers, $a \le b$ and $b \le c$. If $a^2\equiv0\pmod c$ and $b^2\equiv0\pmod c$, and $a$ is the minimal number satisfy $a^2\equiv0\pmod c$, do we always have \$a\...