Question

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

score 734 · Answer 1 · 2016-12-16 13:00:15

734

Views

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

Published on 16 Dec 2016 - 13:00

score 1.2k · Answer 2 · 2016-12-18 01:17:41

1.2k

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there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

Published on 18 Dec 2016 - 1:17

score 65 · Answer 3 · 2016-12-18 19:24:26

65

Views

Are these sequences zero for the greatest of twin, cousin, and prime triples and so on?

Published on 18 Dec 2016 - 19:24

score 186 · Answer 4 · 2026-03-25 15:58:53

186

Views

Is it cheating to use the sign function when sieving for twin primes?

Published on 25 Mar 2026 - 15:58

score 258 · Answer 5 · 2016-12-21 07:01:42

258

Views

Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Published on 21 Dec 2016 - 7:01

score 223 · Answer 6 · 2016-12-23 14:31:08

223

Views

Complete this table of formulas for $_2F_1\big(a,b;c;u) =v $ with algebraic numbers $u,v$ and $a+b=c$?

Published on 23 Dec 2016 - 14:31

score 275 · Answer 7 · 2016-12-23 16:42:00

275

Views

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac14,\frac14;\frac34;-\alpha\right)=\beta\,$?

Published on 23 Dec 2016 - 16:42

score 386 · Answer 8 · 2026-03-25 23:42:31

386

Views

How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

Published on 25 Mar 2026 - 23:42

score 1.9k · Answer 9 · 2026-03-26 02:34:53

1.9k

Views

Collatz Conjecture (3n+1) variant

Published on 26 Mar 2026 - 2:34

score 215 · Answer 10 · 2026-03-26 09:18:07

215

Views

Assuming the Lenstra–Pomerance–Wagstaff conjecture, in what range of values for $p$ would we expect to find the next prime $2^p - 1$?

Published on 26 Mar 2026 - 9:18

score 243 · Answer 11 · 2026-03-26 02:35:59

243

Views

A bound regarding the Collatz conjecture from Wikipedia

Published on 26 Mar 2026 - 2:35

score 588 · Answer 12 · 2017-01-03 05:47:20

588

Views

On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$

Published on 03 Jan 2017 - 5:47

score 216 · Answer 13 · 2017-01-05 13:48:13

216

Views

Connecting $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$ and $\sum\limits_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$

Published on 05 Jan 2017 - 13:48

score 870 · Answer 14 · 2026-03-26 09:17:53

870

Views

When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

Published on 26 Mar 2026 - 9:17

score 117 · Answer 15 · 2026-03-26 09:18:11

117

Views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, does $I(n^2) \geq 5/3$ imply $k=1$?

Published on 26 Mar 2026 - 9:18

List Question

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?

there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

Are these sequences zero for the greatest of twin, cousin, and prime triples and so on?

Is it cheating to use the sign function when sieving for twin primes?

Infinitely $more$ algebraic numbers $\gamma$ and $\delta$ for $_2F_1\left(a,b;\tfrac12;\gamma\right)=\delta$?

Complete this table of formulas for $_2F_1\big(a,b;c;u) =v $ with algebraic numbers $u,v$ and $a+b=c$?

How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac14,\frac14;\frac34;-\alpha\right)=\beta\,$?

How to prove $_2F_1\big(\tfrac16,\tfrac16;\tfrac23;-2^7\phi^9\big)=\large \frac{3}{5^{5/6}}\,\phi^{-1}\,$ with golden ratio $\phi$?

Collatz Conjecture (3n+1) variant

Assuming the Lenstra–Pomerance–Wagstaff conjecture, in what range of values for $p$ would we expect to find the next prime $2^p - 1$?

A bound regarding the Collatz conjecture from Wikipedia

On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$

Connecting $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$ and $\sum\limits_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$

When $p$ is an odd prime, is $(p+2)/p$ an outlaw or an index?

If $q^k n^2$ is an odd perfect number with Euler prime $q$, does $I(n^2) \geq 5/3$ imply $k=1$?

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