Articles of square numbers

Deciding if a number is a square in $\Bbb Z/n\Bbb Z$

I am looking for a systematic way of deciding if a given number is a square in $\Bbb Z/n\Bbb Z$. E.g. is $89$ a square in $\Bbb Z/n\Bbb Z$ for $n\in \{25,33,49\}$? Brute-forcing it would take too long here. How else can we do this?

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one variable)? If not, how would one go about figuring this out?

Is sum of square of primes a square of prime?

I would like to know if it has been proved that : There are no $a$, $b$ and $c$, all prime numbers, such that $a^2 + b^2 = c^2$ There are no $a$, $b$, $c$ and $d$, all prime numbers, such that $a^2 + b^2 + c^2 = d^2$ There are no $a$, $b$, $c$, […]

Prove that $x^2+1$ cannot be a perfect square for any positive integer x?

I started this problem by trying proof by contradiction. I first noted that the problem stated that $x$ had to be a positive integer, and thus $x=0$ could not be a solution. I then assumed that $x^2+1=n^2$ for some integer $n$ other than $1$. From here I have tried various methods, to no avail: Factoring: […]

A mathematical phenomenon regarding perfect squares…

I was working on identifying perfect squares for one of my programs regarding Pythagorean triplet. And I found that for every perfect square if we add its digits recursively until we get a single digit number, e.g. 256 -> 13 -> 4 etc. we get the single digit as either 1,4,7 or 9. Is it […]

Show that $m+3$ and $m^2 + 3m +3$ cannot both be perfect cubes.

Show that $m+3$ and $m^2 + 3m +3$ cannot both be perfect cubes. I’ve done so much algebra on this, but no luck. Tried multiplying, factoring, etc.

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I’m working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we find all sequences of non negative integers ($\forall i\in \mathbb{N} \,\, a_i\in \mathbb{N}$) such that $$\begin{align} \forall i,j,k,l\in\mathbb{N} && i^2+j^2=k^2+l^2 &\Rightarrow a_i^2+a_j^2=a_k^2+a_l^2\end{align}$$ My try: I know […]

Is there an algorithm for writing an integer as a difference of squares?

For example, if we have $36$, is there an algorithm to determine that it may equal $10^2-8^2$? What if we blow up the number to something like $492709612098$? Can it be written as the difference of two squares? If so, how do we know? Also, does it matter if the number in question has more […]

What is the most elementary way of proving a sequence is free of non-trivial squares?

Given the sequence A001921 $$ 0, 7, 104, 1455, 20272, 282359, 3932760, 54776287, 762935264, 10626317415, 148005508552, 2061450802319, 28712305723920, 399910829332567, \dots $$ which obeys the recurrence relation $$ a_0 := 0,\ a_1 := 7, \quad a_{n+2} = 14a_{n+1} – a_n + 6. $$ How can I prove (or, I suppose, disprove) my conjecture that $a_0 = […]

Why is there more room in a square room than there is in a rectangular room when the perimeter is the same in both rooms?

Why is there an extra square foot in a square room with dimensions of $13×13$ and one less square foot in a room with dimensions of $14×12$? The perimeter for both rooms is the same (52 foot). I’m not asking to see formulas giving me the same results I already know – I’d prefer a […]