This tag is for questions involving perfect squares, also known as 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.
Questions tagged [perfect-squares]
21 questions
10
votes
4 answers
Prove $n! +5$ is not a perfect square for $n\in\mathbb{N}$
I have a proof of this simple problem, but I feel that the last step is rather clunky:
For $n=1,2,3,4$ we have $n!+5=6,7,11,29$ respectively, none of which are square. Now assume that $n\geq 5$, then:
$$\begin{aligned}
n! +5 & \;=\; n(n-1)\cdots…
AloneAndConfused
- 1,769
9
votes
2 answers
$(a^2+b^2-ab)(a^2+b^2-2ab)$ is a perfect square
Prove that there are infinitely many integer pairs $(a,b)$ with different value of $\frac ab$ for which $(a^2+b^2-ab)(a^2+b^2-2ab)$ is a perfect square.
techer
- 149
6
votes
2 answers
On the square values of a certain polynomial
Consider the following polynomial on two variables: $$P(a,b)=a^4-4a^3b+6a^2b^2+4ab^3+b^4.$$ Do positive integers $x$ and $y$ exist such that $P(x,y)$ is a perfect square?
I'm aware that this may be actually a very hard problem in disguise, since…
Anonymous Pi
- 1,325
- 8
- 22
4
votes
1 answer
Prove that a 5 digit even number consisting of distinct even digits can't be a perfect square.
I have no idea how to start, except that I know the last two digits must be $24, 04, 84, 64$.
Gerard L.
- 2,591
3
votes
2 answers
Prove that every terms of a sequence defined by a recurrence relation is a perfect square.
I would be happy if you let me know how to tackle this problem.
Thanks.
user488002
- 91
3
votes
0 answers
Polynomial that is a square in every positive integer
Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a polynomial with integer coefficients such that $p(n)$ is a perfect square for every $n\in\mathbb{N}^{*}$. Is it true that exists a polynomial $q$ with rational coefficients such that $p(x)=q(x)^2$?
If…
Terg
- 311
3
votes
0 answers
Determine when $(3^x - y^3)(x^3 - 3^y)$ is a perfect square
Determine all integers $x,y$ such that $f:=(3^x-y^3)(x^3-3^y)$ is a perfect square.
What I have thought: the problem seems a bit hard to begin with. I first tried the situation that $\gcd(x,3)=\gcd(y,3)=1$ and was stuck. It would help a lot if we…
Juggler
- 1,423
2
votes
6 answers
Prove that there exist no natural number k such that $3^k+5^k$ is a square of an integer number?
How to prove that there exist no natural number $k$ such that $3^k+5^k$ is a square of an integer number
user456540
2
votes
1 answer
Sum of digits of perfect square
Prove that the repeated sum of the digits of a perfect square can only be 1,4,7 or 9. Example- 169- Sum-16, Sum of digits of 16 = 7. Please try to explain without log, mods. I am a grade 10th student. Thank You.
Ram Keswani
- 1,013
2
votes
3 answers
Perfect Squares: relatively prime and increasing arithmetic progression
The squares 1, 25, 49 are pairwise relatively prime and in an increasing arithmetic progression. Let $x_1^2, y_1^2, z_1^2$ and $x_2^2, y_2^2, z_2^2$ be the next two triples with this property (so that 49, $z_1^2$, $z_2^2$ are the three smallest…
Annie Smith
- 29
2
votes
1 answer
Permutation such that the sum of a number and its position is a square
We say a positive integer $n$ is good if there exists a permutation $a_1, a_2 . . . a_n$ of $1, 2 . . . n$ such that $k+a_k$ is perfect square for all $1 \le k \le n$. Determine all the good numbers in the set $\{11, 13, 15, 17, 19\}$
I have…
Plato
- 2,372
1
vote
1 answer
Finding multipliers to create nearly perfect squares
Given a very large integer $n$, I wish to find integer multipliers $m$ such that the product is nearly a perfect square. Ideally this should be easy to compute and output a series of improved approximates. We'll define a quality measure as "nearly a…
Diane Wilbor
- 103
0
votes
4 answers
State 2 values of $x$ for which the value of $3x^2+4x-14$ is a perfect square.
State 2 values of $x$ for which the value of $3x^2+4x-14$ is a perfect square.
I can't seem to factor it and I'm really lost. Does using the quadratic equation for it work? Help I need it for grade 10 quadratics test.
0
votes
0 answers
Value of knowing If a number is a perfect square.
I've come up with a method to tell if a number of arbitrary size is a perfect square and was curious..
1) Is there a practical benefit for knowing IF an integer is a perfect square?
2) What is the expected time frame for an integer, N, based on…
Troy W
- 97
0
votes
0 answers
How many squares can form an arithmetic progression?
Related to this question : Find three arithmetic progressions of three square numbers
I am looking for a sequence $$a_1,a_2,\cdots a_n$$ forming an arithmetic progression such that all the $a_j$ are perfect squares, an example with $3$ entries is…
Peter
- 86,576