Question

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

score 163 · Answer 1 · 2024-02-07 16:38:08

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

Published on 07 Feb 2024 - 16:38

score 90 · Answer 2 · 2024-02-07 17:18:51

Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$

score 60 · Answer 3 · 2024-02-08 08:07:28

60

Views

Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs

Published on 08 Feb 2024 - 8:07

score 128 · Answer 4 · 2026-03-26 03:01:34

128

Views

Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)?

Published on 26 Mar 2026 - 3:01

score 64 · Answer 5 · 2026-02-23 06:37:56

64

Views

Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?

Published on 23 Feb 2026 - 6:37

score 33 · Answer 6 · 2024-02-12 15:25:48

33

Views

solution verification | Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$.

Published on 12 Feb 2024 - 15:25

score 89 · Answer 7 · 2026-03-25 01:18:31

89

Views

solution-verification | Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$.

Published on 25 Mar 2026 - 1:18

score 59 · Answer 8 · 2026-03-25 01:17:36

59

Views

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

Published on 25 Mar 2026 - 1:17

score 177 · Answer 9 · 2024-02-13 14:46:52

177

Views

Determine the real numbers $a, b, c, d \in [1,3]$, knowing that the relation $(a + b + c + d)^2 = 3(a^2 + b^2 +c^2 + d^2)$.

Published on 13 Feb 2024 - 14:46

score 117 · Answer 10 · 2024-02-15 16:26:19

117

Views

How to check if something is perfect square or not.

Published on 15 Feb 2024 - 16:26

score 171 · Answer 11 · 2026-03-28 08:51:15

171

Views

Writing 2024 as the sum of 3 and 4 squares

Published on 28 Mar 2026 - 8:51

score 65 · Answer 12 · 2026-03-28 08:50:16

65

Views

Sequence of squares which can't be written as the sum of a smaller non-zero square and twice a triangular number

Published on 28 Mar 2026 - 8:50

score 107 · Answer 13 · 2024-02-16 23:51:46

107

Views

Exists a perfect square that divided by 6 equals a prime number?

Published on 16 Feb 2024 - 23:51

score 177 · Answer 14 · 2024-02-18 15:02:48

177

Views

Find $a$ for which $(a-3)(a-7)$ is a perfect square

Published on 18 Feb 2024 - 15:02

score 87 · Answer 15 · 2024-02-20 10:35:23

87

Views

show that an expression $E(x,y)$ does not depend on x

Published on 20 Feb 2024 - 10:35

List Question

Consider the set $M=\{a^2+3b^2 | a,b \in N\}$ and $n \in N$ so that $25n \in M$. Prove that $2023n \in M$...

Find all the numbers $\overline{abcd}$ (4-digit number in base 10) that check the relationship. $5a^2+5b^2+2c^2+2d^2-2ac-2cd-4ab-4bd-16a+16b-4d+20=0$

Determine the pairs of natural numbers $(x,y)$ for which the relation $2x^2+5y^2-11xy=-25$ occurs

Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)?

Is it possible to define an implicit function for the Kth N such that N(N+1)/2 is a perfect square?

solution verification | Determine the prime natural numbers $p$ for which there exists $x \in Z$ such that $p | x^2+2x+1$ and $p | x^2+8x+16$.

solution-verification | Show that if $a$ and $q$ are natural numbers and the number $(a+\sqrt{q})(a+\sqrt{q+1})$ is rational, then $q=0$.

Prove that there are infinitely many distinct natural numbers $a,b$ for which $\sqrt{a+b}, \sqrt{a-b}$ are simultaneously rational

Determine the real numbers $a, b, c, d \in [1,3]$, knowing that the relation $(a + b + c + d)^2 = 3(a^2 + b^2 +c^2 + d^2)$.

How to check if something is perfect square or not.

Writing 2024 as the sum of 3 and 4 squares

Sequence of squares which can't be written as the sum of a smaller non-zero square and twice a triangular number

Exists a perfect square that divided by 6 equals a prime number?

Find $a$ for which $(a-3)(a-7)$ is a perfect square

show that an expression $E(x,y)$ does not depend on x

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