Modularity and prime number sequence

Question

I tried to solve this modular equation involving first $n$ prime numbers. And it is:

$$2^{3+5+7+11+13+.....+p_{n-3}+p_{n-2}}\equiv p_{n-1}\ \left(\text{mod }p_{n}\right),$$

where $p_{n}$ is the $n$-th prime number.

I couldn't find any solution for this equation until first $300$ primes. Is there any solutions for $n$?

Joffan · Accepted Answer · 2015-02-03T18:47:47.197

1

I found solutions at $p_{n-1}=5813, 29537,$ and $ 44839$ [$p_n = 5821, 29567, $ and $ 44843$ respectively]. We're hitting a fairly random point in the power cycle so I see every reason to expect solutions to continue to appear occasionally.

edited Feb 03 '15 at 18:47

answered Feb 03 '15 at 18:40

Joffan

39,627

Why P(n-1) and not P(n) ? – norak banges Feb 03 '15 at 18:44
@norakbanges I will add those for ease of reference also. – Joffan Feb 03 '15 at 18:45
This maybe very strange.but I'm never doing e-mail in my life.. – norak banges Feb 03 '15 at 19:42

Julián Aguirre · Answer 2 · 2015-02-03T19:02:27.477

1

For what is worth, next solutions $\{n,p_n\}$ are: $$ \{306\,311,4\,353\,467\}\\ \{859\,825,13\,174\,621\}\\ \{1\,700\,098,27\,291\,793\}\\ $$ There are no more with $n\le10^7$.

edited Feb 03 '15 at 19:02

answered Feb 03 '15 at 18:56

Julián Aguirre

76,354

Yes, a $0$ missing. Thanks for noticing. – Julián Aguirre Feb 03 '15 at 19:02

Modularity and prime number sequence

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