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# 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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### Prove that the only natural value of $g$ that makes $8g+4p^2-4p+1$ ($p\in\mathbb{N}$) a perfect square is $g=p$

This problem came up while I was working on a larger problem and I've looked at a few different ways of solving it, the main approach being mathematical induction; however, I've been unable to prove ...
• 68
1 vote
2 answers
47 views

### What is the reason for the ratios of square units not being the same as the ratios of units [closed]

When you have a square with side length 1 yard you get an area of 1 yd$^2$. When you convert the units of this square to feet you get a square with side length 3 and therefore an area of 9 square feet....
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3 votes
3 answers
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### Prove that number $(4n-1)(2n-1)(2n+1)(4n+1)$ is not a perfect square.

the problem Prove that whatever $n$, a non-zero natural number, the number $(4n-1)(2n-1)(2n+1)(4n+1)$ is not a perfect square. my idea I tried working with modular arithmetic in congruences modulo 4, ...
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0 votes
1 answer
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5 votes
3 answers
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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 ...
0 votes
1 answer
110 views

### 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: ...
1 vote
1 answer
67 views

### 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? ...
• 157
3 votes
2 answers
354 views

### 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 ...
• 107
2 votes
2 answers
124 views

### 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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4 votes
3 answers
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0 votes
3 answers
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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 ...
3 votes
2 answers
92 views

### 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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1 vote
1 answer
172 views

### 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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1 vote
2 answers
85 views

• 8,415
0 votes
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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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