Good prime

The term good prime number has different meanings in mathematics. The most common uses relate to comparing a prime number with appropriate means of prime numbers from the surrounding area.

Definition according to Erdős and Straus

The nth prime number is called good if for all pairs of prime numbers and , where goes from 1 to , the following applies: ${\displaystyle p_{n}}$ ${\displaystyle p_{n-i}}$ ${\displaystyle p_{n+i}}$ ${\displaystyle i}$ ${\displaystyle n-1}$

${\displaystyle p_{n}^{2}\;>\;p_{n-i}\cdot p_{n+i}.}$

It can be shown that there are infinitely many good prime numbers. The first of these are

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, ... (episode A028388 in OEIS )

This definition goes back to Paul Erdős and Ernst Gabor Straus .

Examples

Example 1:

We want to test whether 11 is a good prime number.

11 is the fifth prime number . So check: ${\displaystyle 2,3,5,7,\mathbf {11} ,13,17,19,23}$

${\displaystyle 11^{2}=121>7\cdot 13=91}$

${\displaystyle 11^{2}=121>5\cdot 17=85}$

${\displaystyle 11^{2}=121>3\cdot 19=57}$

${\displaystyle 11^{2}=121>2\cdot 23=46}$

So 11 is a good prime number.

Example 2:

We want to test whether 13 is a good prime number.

13 is the sixth prime number . There ${\displaystyle 2,3,5,7,11,\mathbf {13} ,17,19,23,29,31}$

${\displaystyle 13^{2}=169<11\cdot 17=187}$,

does not apply . Hence, 13 is not a good prime number. ${\displaystyle 13=p_{6}^{2}>p_{5}\cdot p_{7}}$

Weaker definition

A prime number is called good if it is greater than the geometric mean of the immediately adjacent pair of prime numbers.

So the nth prime is called good if ${\displaystyle p_{n}}$

${\displaystyle p_{n}^{2}\;>\;p_{n-1}\cdot p_{n+1}}$.

According to this definition, too, there are an infinite number of good prime numbers, the first of which are

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, ... (sequence A046869 in OEIS )

example

The 79 is a good prime number in that sense because

${\displaystyle 79^{2}=6241>73\cdot 83=6059}$.

But it is not a good prime number in the first sense, because it applies to the preceding pair of prime numbers

${\displaystyle 79^{2}=6241<71\cdot 89=6319}$.

Web links

• Eric W. Weisstein : Good Prime . In: MathWorld (English).
• Episode A028388 in OEIS : List of the first 10000 good prime numbers (in the first sense) on On-Line Encyclopedia of Integer Sequences
• Episode A046869 in OEIS : List of the first 10,000 good prime numbers (in the second sense) on the On-Line Encyclopedia of Integer Sequences

Individual evidence

1. ^ Richard Kenneth Guy : Good Primes and the Prime Number Graph. In: Unsolved Problems in Number Theory. 2nd Edition. Springer, New York 1994, p. 32 f, §A14. ( Google books )