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TheRandomGuy
  • Member for 6 years, 11 months
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12 votes
10 answers
3k views

Prove that there are infinitely many integers n such that $4n^2+1$ is divisible by both $13$ and $5$

11 votes
1 answer
2k views

proof - Bézout Coefficients are always relatively prime

11 votes
3 answers
2k views

Solve for integers $x, y, z$ such that $x + y = 1 - z$ and $x^3 + y^3 = 1 - z^2$.

9 votes
3 answers
538 views

Find all integral solutions for the Diophantine Equations $x^4 - x^2y^2 + y^4 = z^2$ and $x^4 + x^2y^2 + y^4 = z^2$.

8 votes
2 answers
2k views

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

8 votes
2 answers
1k views

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$.

8 votes
2 answers
458 views

Prove that the equation $3^k = m^2 + n^2 + 1$ has infinitely many solutions in positive integers.

7 votes
2 answers
346 views

If a, b, c are three natural numbers with $\gcd(a,b,c) = 1$ such that $\frac{1}{a}+\frac{1}{b}=\frac{1}{c}$ then show that $a+b$ is a square.

7 votes
1 answer
1k views

A number 47_ _74 is a multiple of consecutive numbers. Find the numbers.

7 votes
2 answers
359 views

Find all odd $n \in \mathbb{Z}^+$ such that $n\mid 3^n+1$.

6 votes
1 answer
239 views

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers.

6 votes
2 answers
681 views

proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$

5 votes
2 answers
1k views

Show that 13 is the largest prime which divides two consecutive terms of $n^2 + 3$. [duplicate]

5 votes
3 answers
8k views

proof - if $x^2 + y^2 + z^2 = 2xyz$ then $x = y = z = 0$ [duplicate]

5 votes
2 answers
5k views

Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$ [duplicate]

5 votes
2 answers
716 views

Show that $x^2 + y^2 + z^2 = x^3 + y^3 + z^3$ has infinitely many integer solutions.

4 votes
1 answer
2k views

Solving the Diophantine Equation $x^2 - y! = 2001$ and $x^2 - y! = 2016$

4 votes
2 answers
1k views

Prove that if $2n+1$ and $3n+1$ are both perfect squares then $40|n$.

4 votes
0 answers
74 views

Show that any positive rational number can be expressed as $\frac{a^3+b^3}{c^3+d^3}$. [duplicate]

4 votes
4 answers
448 views

Prove that the diophantine equation $x^2 + (x+1)^2 = y^2$ has infinitely many solutions in positive integers.

4 votes
2 answers
462 views

Prove that however one selects $55$ integers $1 ≤ x_1 < x_2 < x_3 < ... < x_{55} ≤ 100$, there will be some two that differ by 9, 10, 12 and 13.

4 votes
1 answer
193 views

proof - $\forall a, b \in \mathbb{Z}^+, a \neq b, \exists \text{ infinite } n \in \mathbb{Z}^. \gcd(a+n, b+n) = 1$

4 votes
1 answer
323 views

Show that there exist no $a, b, c \in \mathbb Z^+$ such that $a^3 + 2b^3 = 4c^3$

4 votes
1 answer
2k views

Show that if $x,y,z$ are positive integers, then $(xy + 1)(yz + 1)(zx + 1)$ is a perfect square iff $xy +1, yz +1, zx+1$ are all perfect squares.

4 votes
2 answers
214 views

Find infinitely many pairs of integers $a$ and $b$ with $1 < a < b$, so that $ab$ exactly divides $a^2 +b^2 −1$

4 votes
1 answer
132 views

Several different positive integers lie strictly between two successive squares. Prove that their pairwise products are also different.

3 votes
3 answers
910 views

If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.

3 votes
1 answer
505 views

Solve the equation $x^3 + 117y^3 = 5$ over the integers.

3 votes
1 answer
122 views

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

3 votes
2 answers
915 views

Twin Primes, their Arithmetic Means and some properties.