From Wikipedia, the free encyclopedia
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:
5 ,
11 ,
17 ,
29 ,
37 ,
41 ,
53 ,
59 ,
67 ,
71 ,
97 ,
101 ,
127 ,
149 ,
179 ,
191 ,
223 ,
227 ,
251 ,
257 ,
269 ,
307 ,
311 ,
331 ,
347 ,
419 ,
431 ,
541 ,
557 ,
563 ,
569 ,
587 ,
593 ,
599 ,
641 ,
727 ,
733 ,
739 ,
809 ,
821 ,
853 ,
929 ,
937 ,
967 (sequence
A028388 in the
OEIS ).
An alternative version takes only i = 1 in the definition. With that there are more good primes:
5 ,
11 ,
17 ,
29 ,
37 ,
41 ,
53 ,
59 ,
67 ,
71 ,
79 ,
97 ,
101 ,
107 ,
127 ,
137 ,
149 ,
157 ,
163 ,
173 ,
179 ,
191 ,
197 ,
211 ,
223 ,
227 ,
239 ,
251 ,
257 ,
263 ,
269 ,
277 ,
281 ,
307 ,
311 ,
331 ,
347 ,
367 ,
373 ,
379 ,
397 ,
419 ,
431 ,
439 ,
457 ,
461 ,
479 ,
487 ,
499 ,
521 ,
541 ,
557 ,
563 ,
569 ,
587 ,
593 ,
599 ,
607 ,
613 ,
617 ,
631 ,
641 ,
653 ,
659 ,
673 ,
701 ,
719 ,
727 ,
733 ,
739 ,
751 ,
757 ,
769 ,
787 ,
809 ,
821 ,
827 ,
853 ,
857 ,
877 ,
881 ,
907 ,
929 ,
937 ,
947 ,
967 ,
977 ,
991 (sequence
A046869 in the
OEIS ).
References
By formula By integer sequence By property
Base -dependent Patterns
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