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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.
That is, good prime satisfies the inequality
p
n
2
>
p
n
−
i
⋅
p
n
+
i
{\displaystyle p_{n}^{2}>p_{n-i}\cdot p_{n+i}}
for all 1 ≤ i ≤ n −1, where pk is the k th prime.
Example: the first primes are 2, 3, 5, 7 and 11. Since for 5 both the conditions
5
2
>
3
⋅
7
{\displaystyle 5^{2}>3\cdot 7}
5
2
>
2
⋅
11
{\displaystyle 5^{2}>2\cdot 11}
are fulfilled, 5 is a good prime.
There are infinitely many good primes.[1] The first good primes are:
).
An alternative version takes only i = 1 in the definition. With that there are more good primes:
).
References
By formula By integer sequence By property Base -dependentPatterns
Twin (p , p + 2 )
Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, … )
Triplet (p , p + 2 or p + 4, p + 6 )
Quadruplet (p , p + 2, p + 6, p + 8 )
k -tuple
Cousin (p , p + 4 )
Sexy (p , p + 6 )
Chen
Sophie Germain/Safe (p , 2p + 1 )
Cunningham (p , 2p ± 1, 4p ± 3, 8p ± 7, ... )
Arithmetic progression (p + a·n , n = 0, 1, 2, 3, ... )
Balanced (consecutive p − n , p , p + n )
By size Complex numbers Composite numbers Related topics First 60 primes