Question

How can I prove this inequality? $\frac{p_k}{p_{k+1} + 1} < \frac{\ln(\ln(Primorial(k)))}{\ln(\ln(Primorial(k+1)))}$

score 136 · Answer 1 · 2026-03-25 18:49:33

136

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How can I prove this inequality? $\frac{p_k}{p_{k+1} + 1} < \frac{\ln(\ln(Primorial(k)))}{\ln(\ln(Primorial(k+1)))}$

Published on 25 Mar 2026 - 18:49

score 253 · Answer 2 · 2026-03-25 18:47:10

253

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Contradiction between OEIS and factordb.com

Published on 25 Mar 2026 - 18:47

score 116 · Answer 3 · 2026-03-25 18:43:45

116

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Reference request: The primorial

Published on 25 Mar 2026 - 18:43

score 357 · Answer 4 · 2026-03-25 18:50:13

357

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I'm trying to generalize some simple results about $ 2 ^ n $. It's useful to write them in primoradic (see stub OEIS).

Published on 25 Mar 2026 - 18:50

score 181 · Answer 5 · 2026-03-25 18:50:12

181

Views

I'm looking for a proof for $\frac 14=\sum_{k=2}^{\infty}\frac{\left(\frac{p_k-1}{2}\right)}{P(p_k)}$

Published on 25 Mar 2026 - 18:50

score 157 · Answer 6 · 2026-03-25 18:49:42

157

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What is the primorial development of e ?(the notion of "primorial development" is defined in the body)

Published on 25 Mar 2026 - 18:49

score 111 · Answer 7 · 2026-03-25 18:48:09

111

Views

A divisibility rule for 7 in primoradic with an historical example. I'm looking for other examples of the same type.

Published on 25 Mar 2026 - 18:48

score 210 · Answer 8 · 2026-03-25 18:48:34

210

Views

A result about Fermat's numbers. Is my proof correct? Is that result useful ?Can we generalize that result?

Published on 25 Mar 2026 - 18:48

score 141 · Answer 9 · 2026-03-25 18:48:39

141

Views

Fermat numbers are congruent to $17$ or $47$ modulo $210$. Can we generalize this result?

Published on 25 Mar 2026 - 18:48

score 116 · Answer 10 · 2026-03-25 18:50:18

116

Views

Showing existence in congruence $q\equiv d^k+\frac{mp_n\#}d\pmod{p_n\#}$

Published on 25 Mar 2026 - 18:50

score 88 · Answer 11 · 2026-03-25 18:52:07

88

Views

Heuristics for Square-free Proof?

Published on 25 Mar 2026 - 18:52

score 51 · Answer 12 · 2026-03-25 18:49:34

51

Views

Combining disjoint congruences: $q\equiv \prod_ip_{u_i}^{r_{u_i}}\mod d$ and $q\equiv \prod_ip_{u_i}^{s_{u_i}}\mod p_{n+1}$

Published on 25 Mar 2026 - 18:49

score 42 · Answer 13 · 2026-03-25 18:49:51

42

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An open problem concerns primorials plus one having square factors. What about primorials squared plus one?

Published on 25 Mar 2026 - 18:49

score 4k · Answer 14 · 2026-03-25 18:47:02

4k

Views

Why does the primorial $23\#$ come up so often in long prime arithmetic progressions?

Published on 25 Mar 2026 - 18:47

score 70 · Answer 15 · 2026-03-25 18:49:07

70

Views

Minimum value made of the reciprocals of the first $n$ primes

Published on 25 Mar 2026 - 18:49

List Question

How can I prove this inequality? $\frac{p_k}{p_{k+1} + 1} < \frac{\ln(\ln(Primorial(k)))}{\ln(\ln(Primorial(k+1)))}$

Contradiction between OEIS and factordb.com

Reference request: The primorial

I'm trying to generalize some simple results about $ 2 ^ n $. It's useful to write them in primoradic (see stub OEIS).

I'm looking for a proof for $\frac 14=\sum_{k=2}^{\infty}\frac{\left(\frac{p_k-1}{2}\right)}{P(p_k)}$

What is the primorial development of e ?(the notion of "primorial development" is defined in the body)

A divisibility rule for 7 in primoradic with an historical example. I'm looking for other examples of the same type.

A result about Fermat's numbers. Is my proof correct? Is that result useful ?Can we generalize that result?

Fermat numbers are congruent to $17$ or $47$ modulo $210$. Can we generalize this result?

Showing existence in congruence $q\equiv d^k+\frac{mp_n\#}d\pmod{p_n\#}$

Heuristics for Square-free Proof?

Combining disjoint congruences: $q\equiv \prod_ip_{u_i}^{r_{u_i}}\mod d$ and $q\equiv \prod_ip_{u_i}^{s_{u_i}}\mod p_{n+1}$

An open problem concerns primorials plus one having square factors. What about primorials squared plus one?

Why does the primorial $23\#$ come up so often in long prime arithmetic progressions?

Minimum value made of the reciprocals of the first $n$ primes

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