A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes.
A good prime satisfies the inequality
\( p_{n}^{2}>p_{{(n-i)}}\cdot p_{{(n+i)}} \)
for all 1 ≤ i ≤ n−1. pn is the nth prime.
Example: The first primes are 2, 3, 5, 7 and 11. As for 5 both possible conditions
\( 5^{2}>3\cdot 7 \)
\( 5^{2}>2\cdot 11 \)
are fulfilled, 5 is a good prime.
There are infinitely many good primes.[1] The first few good primes are
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149 (sequence A028388 in the OEIS).
References
Weisstein, Eric W. "Good Prime". MathWorld.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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