Question

For any odd integer $n > 2$, show that there isn't any positive integer $x$, such that $x^n + (x+1)^n = (x+2)^n$.

score 69 · Answer 1 · 2018-07-18 17:40:46

69

Views

For any odd integer $n > 2$, show that there isn't any positive integer $x$, such that $x^n + (x+1)^n = (x+2)^n$.

Published on 18 Jul 2018 - 17:40

score 1.1k · Answer 2 · 2026-03-26 16:06:25

1.1k

Views

Combinatorial proof of Negative Binomial Identity $(1+x)^{-n}= \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k$

Published on 26 Mar 2026 - 16:06

score 99 · Answer 3 · 2018-07-19 14:28:22

99

Views

Prove that $\binom{2n}{n}\equiv (-1)^n \pmod{(2n+1)}$ if and only if $2n+1$ is a prime number.

Published on 19 Jul 2018 - 14:28

score 221 · Answer 4 · 2018-07-19 17:36:06

221

Views

Compute $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$ using Fourier series method.

Published on 19 Jul 2018 - 17:36

score 53 · Answer 5 · 2026-03-25 06:55:05

53

Views

What composite number can be represented as $\binom{n}{k}$ for $k\neq1$ or $n-1$

Published on 25 Mar 2026 - 6:55

score 294 · Answer 6 · 2026-03-26 14:34:03

294

Views

number of different ways to represent a positive integer as a binomial coefficient

Published on 26 Mar 2026 - 14:34

score 64 · Answer 7 · 2018-07-20 02:19:01

64

Views

Having trouble with a Binomial proof by mathematical induction question: $\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$

Published on 20 Jul 2018 - 2:19

score 98 · Answer 8 · 2026-03-25 14:25:25

98

Views

$\lim_{n \to \infty} \sum_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}$?

Published on 25 Mar 2026 - 14:25

score 67 · Answer 9 · 2026-03-25 12:50:55

67

Views

Is it true that $\sum_{j=k}^{2k} {2k \choose j}\big(\frac{1}{2}\big)^{2k}-\frac{1}{2}=\frac{1}{2}\Big({2k \choose k}\big(\frac{1}{2}\big)^{2k}\Big)$?

Published on 25 Mar 2026 - 12:50

score 2.4k · Answer 10 · 2018-07-23 13:09:41

2.4k

Views

Prove $\binom{2n}{n}~=~\sum_{k=0}^n \binom{n}{k}^2$

Published on 23 Jul 2018 - 13:09

score 247 · Answer 11 · 2026-03-25 04:40:07

247

Views

What's the probability of completing the illustrated "binomial walk" without ever visiting a node above the baseline?

Published on 25 Mar 2026 - 4:40

score 442 · Answer 12 · 2018-07-24 05:48:18

442

Views

Binomial Coefficient Identity Conjecture

Published on 24 Jul 2018 - 5:48

score 115 · Answer 13 · 2018-07-24 08:01:55

115

Views

algebraic derivation of sum of binomial coefficients

Published on 24 Jul 2018 - 8:01

score 642 · Answer 14 · 2026-03-25 06:05:21

642

Views

entropy of the sum of binomial distributions

Published on 25 Mar 2026 - 6:05

score 335 · Answer 15 · 2026-03-23 01:19:41

335

Views

Maximum $n$ such that ${n \choose k} \,2^{1 - {k \choose 2}} < 1$ (where $k$ is a constant)

Published on 23 Mar 2026 - 1:19

List Question

For any odd integer $n > 2$, show that there isn't any positive integer $x$, such that $x^n + (x+1)^n = (x+2)^n$.

Combinatorial proof of Negative Binomial Identity $(1+x)^{-n}= \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k$

Prove that $\binom{2n}{n}\equiv (-1)^n \pmod{(2n+1)}$ if and only if $2n+1$ is a prime number.

Compute $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$ using Fourier series method.

What composite number can be represented as $\binom{n}{k}$ for $k\neq1$ or $n-1$

number of different ways to represent a positive integer as a binomial coefficient

Having trouble with a Binomial proof by mathematical induction question: $\sum_{j=3}^n \binom{j-1}{2} = \binom{n}{3}$

$\lim_{n \to \infty} \sum_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}$?

Is it true that $\sum_{j=k}^{2k} {2k \choose j}\big(\frac{1}{2}\big)^{2k}-\frac{1}{2}=\frac{1}{2}\Big({2k \choose k}\big(\frac{1}{2}\big)^{2k}\Big)$?

Prove $\binom{2n}{n}~=~\sum_{k=0}^n \binom{n}{k}^2$

What's the probability of completing the illustrated "binomial walk" without ever visiting a node above the baseline?

Binomial Coefficient Identity Conjecture

algebraic derivation of sum of binomial coefficients

entropy of the sum of binomial distributions

Maximum $n$ such that ${n \choose k} \,2^{1 - {k \choose 2}} < 1$ (where $k$ is a constant)

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