Pillaic prime number

In number theory, one is Pillaische prime a prime number , for which a positive integer exists, so that the faculty of , ie , is smaller by one than a multiple of a prime number . However, the prime number itself must not be one greater than a multiple of . In other words: ${\displaystyle p}$ ${\displaystyle n>0}$ ${\displaystyle n}$ ${\displaystyle n!}$ ${\displaystyle p}$ ${\displaystyle n}$

There is one with and it must be for everyone . ${\displaystyle k_{1}\in \mathbb {N} }$ ${\displaystyle n!+1=k_{1}\cdot p}$ ${\displaystyle p-1\not =k_{2}\cdot n}$ ${\displaystyle k_{2}\in \mathbb {N} }$

Written with congruences this means:

It must   and   apply. ${\displaystyle n!+1\equiv 0{\pmod {p}}}$ ${\displaystyle p\not \equiv 1{\pmod {n}}}$

The corresponding numbers are called EHS numbers . ${\displaystyle n\in \mathbb {N} }$

The Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai , who first looked at these numbers by wondering whether it is true that every prime divisor of is of the form . ${\displaystyle p\in \mathbb {P} }$ ${\displaystyle n!+1}$ ${\displaystyle p=k\cdot n+1}$

Examples

• The number is a Pillai pim number because: ${\displaystyle p=137}$

With and applies: and it actually applies to everyone . ${\displaystyle n=16}$ ${\displaystyle k=152721094073}$ ${\displaystyle n!+1=16!+1=20922789888001=152721094073\cdot 137=k_{1}\cdot p}$ ${\displaystyle p-1=137-1=136\not =k_{2}\cdot 16=k_{2}\cdot n}$ ${\displaystyle k_{2}\in \mathbb {N} }$

• The first Pillai primes are the following:

23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499, 503, 521, 557, 563, 569, 571, 577, 593, 599, 601, 607, 613, 619, 631, 641, 647, 661, 673, ... (sequence A063980 in OEIS )

• The first EHS numbers are the following:

8, 9, 13, 14, 15, 16, 17, 18, 19, 22, ...

The smallest (there are several) belonging to the above list are the following: ${\displaystyle p\in \mathbb {P} }$

61, 71, 83, 23, 59, 61, 661, 23, 71, 521, ...

Example: The EHS number and the prime number can be found in the 7th position in the two lists above . Indeed it is and it is indeed for everyone . ${\displaystyle n=17}$ ${\displaystyle p=661}$ ${\displaystyle n!+1=17!+1=355687428096001=538105034941\cdot 661=k_{1}\cdot p}$ ${\displaystyle p-1=661-1=660\not =k_{2}\cdot 17=k_{2}\cdot n}$ ${\displaystyle k_{2}\in \mathbb {N} }$

properties

• There are infinitely many Pillai primes.
• There are an infinite number of EHS numbers.

Unsolved problems

The following unsolved issues are raised in:

• Let the number of prime numbers be less than or equal and the number of Pillai prime numbers less than or equal . ${\displaystyle \pi (x)}$ ${\displaystyle x}$ ${\displaystyle \pi ({\mathcal {P}},x)}$ ${\displaystyle x}$

Is ? ${\displaystyle \lim _{x\to \infty }{\frac {\pi ({\mathcal {P}},x)}{\pi (x)}}=1}$

• Let the number of EHS numbers be less than or equal . ${\displaystyle f(x)}$ ${\displaystyle x}$

Exists ? ${\displaystyle \lim _{x\to \infty }{\frac {f(x)}{x}}}$

If so, what value is this limit going against ?

It is and and and and . ${\displaystyle {\frac {f(100)}{100}}\approx 5{,}5}$ ${\displaystyle {\frac {f(200)}{200}}\approx 5{,}25}$ ${\displaystyle {\frac {f(300)}{300}}\approx 5{,}7}$ ${\displaystyle {\frac {f(400)}{400}}\approx 5{,}45}$ ${\displaystyle {\frac {f(500)}{500}}\approx 4{,}98}$

It could be that the Limes is, if it exists. ${\displaystyle 0{,}5}$

Individual evidence

1. ↑ ^{a b c d e} G. E. Hardy, MV Subbarao: A Modified Problem of Pillai and Some Related Questions. The American Mathematical Monthly 109 (6), 2002, pp. 554-559 , accessed June 13, 2018 . 

Web links

• Pillai prime . In: PlanetMath . (English)
• Chris K. Caldwell: Pillai prime. The Prime Glossary, accessed June 13, 2018 . 

swell

• GE Hardy, MV Subbarao: A Modified Problem of Pillai and Some Related Questions . In: The American Mathematical Monthly . tape 109 , no. 6 , 2002, pp. 554-559 .
• RK Guy: Unsolved Problems in Number Theory . 3. Edition. Springer-Verlag, New York 2004, ISBN 0-387-20860-7 .