All Questions
12
questions
4
votes
1
answer
84
views
When is $3136 - 6272 n + 3136 n^2 + n^{14}$ a perfect square?
When is $3136 - 6272 n + 3136 n^2 + n^{14} =z^2$ for $z \in \mathbb N$?
Here was my approach: Consider $3136 - 6272 n + 3136 n^2 + n^{14} \pmod 5$. Since any integral square must be equivalent to 0, 1,...
- 972
0
votes
0
answers
105
views
Perfect square equation $12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2$
Well, I have the following function:
$$\text{y}\left(x\right):=12\alpha^2\cdot x^3+12\alpha\cdot x^2+12\alpha\left(1-\alpha\right)\cdot x+\left(2-3\alpha\right)^2\tag1$$
Where $\alpha\in\mathbb{N}$.
...
- 28.7k
0
votes
1
answer
91
views
Is the polynomial $p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$?
My initial question in the present post is pretty basic:
Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$, and ...
- 8,010
4
votes
2
answers
575
views
Does a generalized difference of powers formula exist?
The identity
$$
\left(\frac{n+1}{2}\right)^2-\left(\frac{n-1}{2}\right)^2=n
$$
can be used to represent any number as difference of two squares. (Note that this formula gives integer values when $n$ ...
- 3,187
2
votes
2
answers
135
views
Perfect Square With Two Integer Variables
I am trying to solve a number theory problem in general form. However, I got stuck in the following step:
$a,b,n \in \mathbb Z^{+}$ for which values of $n$, this equation is solvable $\frac{(n+1)(n+2a)...
- 49
0
votes
3
answers
77
views
Complete square to show that $(x-2y)^4+64xy+16 \geq 0$
I want to prove that
$f(x,y) = (x-2y)^4+64xy+16 \geq 0$
Using only algebraic methods (I guess it can be solved via completing the square).
I didn't manage to do so. Any help will be appreciated.
...
- 289
2
votes
3
answers
213
views
Generalizing $\,r(n^2) = r(n)^2,\,$ for $\,r(n) := $ reverse the digits of $n$
I'm assuming this theorem was found by someone else before, but I found this relationship between square numbers of 3 digits or less. The theorem is this: If you reverse the digits in a square number, ...
- 139
2
votes
3
answers
97
views
$f(n)+f(m) = q^2$ always has a solution
Prove or disprove: Let $f$ be a non-constant polynomial with nonnegative integer coefficients. Then there exist $m,n \in \mathbb{N}$ such that $f(n)+f(m)$ is a perfect square.
I'm just posting this ...
- 1,567
6
votes
2
answers
137
views
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 ...
- 1,297
0
votes
0
answers
50
views
Number of solutions of that congruence?
We consider $L=\ \{x\in (\mathbb{Z}/2^3 \mathbb{Z})^{\times} \ \vert \ x^2 \equiv 1 \pmod{2^3}\}$. I want to determine $Card(L)$.
If the group was the cyclic the answer would be $Card(L)=\gcd(4,2)=...
- 3,330
3
votes
1
answer
127
views
Find an explicit formula for a square integer between $4n^3$ and $4n^3 + 4n^2$
Given any positive integer $n>1$, it is fairly straightforward to prove that there exists at least one (and perhaps many) square integers lying strictly between $4n^3$ and $4n^3 + 4n^2$. (Proof: ...
- 23.7k
2
votes
1
answer
562
views
Polynomial whose only values are squares
Given a polynomial $ P \in \Bbb Z [X] $ such that, $ P (x)$ is the square of an integer for all integers x, is $ P $ necessarily of the form $ P (x)= Q (x)^2$ with $ Q \in \Bbb Z [X]$?
- 14.4k