Question

How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

score 477 · Answer 1 · 2026-05-10 19:03:44

477

Views

How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

Published on 10 May 2026 - 19:03

score 180 · Answer 2 · 2026-05-10 17:00:19

180

Views

Primality test using the sum of 2 squares representation of a prime $p^2= a^2+b^2$

Published on 10 May 2026 - 17:00

score 331 · Answer 3 · 2026-04-14 00:26:54

331

Views

Prove for any positive integer $n$, $(4n)!$ is divisible by $2^{3n}\cdot 3^n$

Published on 14 Apr 2026 - 0:26

score 192 · Answer 4 · 2026-04-11 19:36:19

192

Views

Check divisibility by using $\gcd$.

Published on 11 Apr 2026 - 19:36

score 127 · Answer 5 · 2026-04-15 18:09:39

127

Views

Prove that there are no natural numbers $x$ whose digits are $0$ or $2$, such that $x$ is a perfect square.

Published on 15 Apr 2026 - 18:09

score 1.1k · Answer 6 · 2026-05-15 02:44:04

1.1k

Views

Primes congruent to a mod n.

Published on 15 May 2026 - 2:44

score 84 · Answer 7 · 2026-04-11 23:38:53

84

Views

Sum and Product of Root

Published on 11 Apr 2026 - 23:38

score 228 · Answer 8 · 2026-04-14 17:53:04

228

Views

Roots of unity (group theory)

Published on 14 Apr 2026 - 17:53

score 1k · Answer 9 · 2026-05-15 10:48:41

1k

Views

Given 20 distinct positive integers, not greater than 70. Show that among their pairwise differences, at least four are equal.

Published on 15 May 2026 - 10:48

score 71 · Answer 10 · 2026-05-04 12:24:56

71

Views

Proof Verification: Prove that $\gcd((a+b)^m, (a-b)^m)\leq2^m$ for relatively prime $a$ and $b$.

Published on 04 May 2026 - 12:24

score 62 · Answer 11 · 2026-04-14 20:53:08

62

Views

Number of elements in shifted powersets of positive integers

Published on 14 Apr 2026 - 20:53

score 584 · Answer 12 · 2026-04-16 15:35:35

584

Views

How do we prove that $[n+1/2]+[n+2/4]+[n+4/8]+[n+8/16] ...=n$?

Published on 16 Apr 2026 - 15:35

score 132 · Answer 13 · 2026-04-16 04:27:51

132

Views

Find all integers $x, y, u, v$ for which holds: $x^2 + y^2 = 3(u^2 + v^2)$.

Published on 16 Apr 2026 - 4:27

score 3.7k · Answer 14 · 2026-04-13 20:07:13

3.7k

Views

Prove that if $ab$ is a perfect square and $GCD(a,b)=1$, then $a$ and $b$ are perfect squares

Published on 13 Apr 2026 - 20:07

score 111 · Answer 15 · 2026-04-13 19:40:54

111

Views

show that the remainder upon dividing $a^n-1$ by $a^m-1$ is $a^r-1$

Published on 13 Apr 2026 - 19:40

List Question

How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

Primality test using the sum of 2 squares representation of a prime $p^2= a^2+b^2$

Prove for any positive integer $n$, $(4n)!$ is divisible by $2^{3n}\cdot 3^n$

Check divisibility by using $\gcd$.

Prove that there are no natural numbers $x$ whose digits are $0$ or $2$, such that $x$ is a perfect square.

Primes congruent to a mod n.

Sum and Product of Root

Roots of unity (group theory)

Given 20 distinct positive integers, not greater than 70. Show that among their pairwise differences, at least four are equal.

Proof Verification: Prove that $\gcd((a+b)^m, (a-b)^m)\leq2^m$ for relatively prime $a$ and $b$.

Number of elements in shifted powersets of positive integers

How do we prove that $[n+1/2]+[n+2/4]+[n+4/8]+[n+8/16] ...=n$?

Find all integers $x, y, u, v$ for which holds: $x^2 + y^2 = 3(u^2 + v^2)$.

Prove that if $ab$ is a perfect square and $GCD(a,b)=1$, then $a$ and $b$ are perfect squares

show that the remainder upon dividing $a^n-1$ by $a^m-1$ is $a^r-1$

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