Question

Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$

score 117 · Answer 1 · 2023-09-20 14:29:26

117

Views

Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$

Published on 20 Sep 2023 - 14:29

score 94 · Answer 2 · 2023-09-23 18:21:37

94

Views

Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.

Published on 23 Sep 2023 - 18:21

score 300 · Answer 3 · 2026-03-26 04:30:36

300

Views

Can $!1+!2+!3+\cdots+!n$ be a perfect power?

Published on 26 Mar 2026 - 4:30

score 327 · Answer 4 · 2026-03-29 03:36:41

327

Views

Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?

Published on 29 Mar 2026 - 3:36

score 152 · Answer 5 · 2023-09-28 16:27:55

152

Views

Prove $a=b$ if $4p^{2a}+4p^b+1$ is a perfect square

Published on 28 Sep 2023 - 16:27

score 103 · Answer 6 · 2023-09-29 14:43:07

103

Views

If $x>y\ge1$ are integers, with $x$ even and $y$ odd, and $x^4+4x^3y-4xy^3-4y^4$ is a square, must $y \mid x$?

Published on 29 Sep 2023 - 14:43

score 93 · Answer 7 · 2023-09-30 13:09:50

93

Views

Is it possible for three distinct positive integers to exist such that sum and difference of every two is a perfect square?

Published on 30 Sep 2023 - 13:09

score 59 · Answer 8 · 2023-10-01 01:31:22

59

Views

Is there a function that can cancel a square of a number, preserving its initial value?

Published on 01 Oct 2023 - 1:31

score 178 · Answer 9 · 2023-10-04 18:13:30

178

Views

$3^{2n}+3n^2+7$, a perfect square?

Published on 04 Oct 2023 - 18:13

score 88 · Answer 10 · 2023-10-05 18:13:16

88

Views

Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{ 3}}$ there is at least one irrational one.

Published on 05 Oct 2023 - 18:13

score 126 · Answer 11 · 2023-10-07 04:26:06

126

Views

Every non-square number is given by the sequence $a(n) = n + \Big\lfloor \frac 12 + \sqrt n\Big\rfloor$

Published on 07 Oct 2023 - 4:26

score 214 · Answer 12 · 2026-02-24 03:50:43

214

Views

Determine all ordered pairs $(m, k)$ for which $f(m, km)$ is a perfect square

Published on 24 Feb 2026 - 3:50

score 230 · Answer 13 · 2023-10-18 15:38:23

230

Views

Prove that $N_n = n(n-1)/2$.

Published on 18 Oct 2023 - 15:38

score 48 · Answer 14 · 2023-10-23 19:58:02

48

Views

Explain the logic in this math equation

Published on 23 Oct 2023 - 19:58

score 47 · Answer 15 · 2023-10-25 09:25:16

47

Views

When is a proth number a perfect square?

Published on 25 Oct 2023 - 9:25

List Question

Show that if the numbers $\sqrt{a^2+b+c+1}$, $\sqrt{b^2+c+a+1}$ and $\sqrt{c^2+a+b+1 }$ are rational, then $a = b = c$

Solve the equation $3x^2+2y^2-4xy+x-2=0$ in the set of integer numbers.

Can $!1+!2+!3+\cdots+!n$ be a perfect power?

Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?

Prove $a=b$ if $4p^{2a}+4p^b+1$ is a perfect square

If $x>y\ge1$ are integers, with $x$ even and $y$ odd, and $x^4+4x^3y-4xy^3-4y^4$ is a square, must $y \mid x$?

Is it possible for three distinct positive integers to exist such that sum and difference of every two is a perfect square?

Is there a function that can cancel a square of a number, preserving its initial value?

$3^{2n}+3n^2+7$, a perfect square?

Show that among the numbers $\sqrt{\frac{pq-2}{3}}$, $\sqrt{\frac{qr-2}{3}}$ and $\sqrt{\frac{pr-2}{ 3}}$ there is at least one irrational one.

Every non-square number is given by the sequence $a(n) = n + \Big\lfloor \frac 12 + \sqrt n\Big\rfloor$

Determine all ordered pairs $(m, k)$ for which $f(m, km)$ is a perfect square

Prove that $N_n = n(n-1)/2$.

Explain the logic in this math equation

When is a proth number a perfect square?

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