User TheRandomGuy

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 $5$

asked Mar 19, 2016 at 14:09

11 votes

1 answer

2k views

proof - Bézout Coefficients are always relatively prime

number-theory elementary-number-theory proof-verification

asked Feb 28, 2016 at 8:45

11 votes

3 answers

2k views

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

number-theory elementary-number-theory polynomials diophantine-equations cubics

asked May 30, 2016 at 9:53

9 votes

3 answers

538 views

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

number-theory diophantine-equations elliptic-curves

asked Apr 26, 2016 at 10:05

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 $x, y \in \mathbb{Z}^+$.

number-theory elementary-number-theory diophantine-equations

asked Apr 22, 2016 at 14:32

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 $x = y$.

number-theory elementary-number-theory square-numbers

asked Jun 16, 2016 at 10:40

8 votes

2 answers

458 views

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

elementary-number-theory diophantine-equations

asked Mar 29, 2016 at 13:47

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.

number-theory elementary-number-theory divisibility gcd-and-lcm square-numbers

asked Mar 18, 2016 at 4:55

7 votes

1 answer

1k views

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

number-theory elementary-number-theory arithmetic cryptarithm

asked Mar 14, 2016 at 10:46

7 votes

2 answers

359 views

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

elementary-number-theory divisibility

asked May 30, 2016 at 13:12

6 votes

1 answer

239 views

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

number-theory elementary-number-theory diophantine-equations

asked Apr 13, 2016 at 10:08

6 votes

2 answers

681 views

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

number-theory elementary-number-theory summation factorial perfect-powers

asked Mar 18, 2016 at 6:22

5 votes

2 answers

1k views

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

number-theory elementary-number-theory

asked Mar 23, 2016 at 17:24

5 votes

3 answers

8k views

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

number-theory elementary-number-theory diophantine-equations

asked Feb 20, 2016 at 11:56

5 votes

2 answers

5k views

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

elementary-number-theory divisibility gcd-and-lcm

asked Mar 3, 2016 at 13:57

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.

number-theory elementary-number-theory diophantine-equations sums-of-squares

asked Apr 7, 2016 at 14:33

4 votes

1 answer

2k views

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

number-theory elementary-number-theory diophantine-equations

asked Apr 20, 2016 at 11:43

4 votes

2 answers

1k views

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

number-theory elementary-number-theory proof-verification alternative-proof

asked May 7, 2016 at 15:49

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]

number-theory elementary-number-theory

asked May 25, 2016 at 14:31

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.

number-theory elementary-number-theory diophantine-equations

asked May 28, 2016 at 15:47

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.

combinatorics pigeonhole-principle

asked Dec 9, 2016 at 11:13

4 votes

1 answer

193 views

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

elementary-number-theory gcd-and-lcm

asked Mar 17, 2016 at 9:18

4 votes

1 answer

323 views

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

number-theory elementary-number-theory proof-verification diophantine-equations

asked Feb 20, 2016 at 10:05

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.

number-theory elementary-number-theory

asked Mar 19, 2016 at 10:08

4 votes

2 answers

214 views

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

number-theory elementary-number-theory proof-verification alternative-proof

asked Mar 20, 2016 at 10:06

4 votes

1 answer

132 views

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

number-theory elementary-number-theory

asked Apr 4, 2016 at 10:22

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 $n^2 = x^2 + y^2 + z^2$.

number-theory elementary-number-theory sums-of-squares

asked Apr 6, 2016 at 11:33

3 votes

1 answer

505 views

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

number-theory elementary-number-theory diophantine-equations

asked Mar 30, 2016 at 8:50

3 votes

1 answer

122 views

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

number-theory elementary-number-theory diophantine-equations

asked Apr 1, 2016 at 14:32

3 votes

2 answers

915 views

Twin Primes, their Arithmetic Means and some properties.

number-theory elementary-number-theory prime-numbers prime-twins

asked Mar 24, 2016 at 16:06

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79 Questions

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

proof - Bézout Coefficients are always relatively prime

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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.

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

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

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 $n^2 = x^2 + y^2 + z^2$.

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

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

Twin Primes, their Arithmetic Means and some properties.