Search a number
good primes
A prime  $p_n$  is said to be good if it  $p_n^2 > p_{n-i}\cdot p_{n+i}$  for every  $1\le i<n$.

For example,  $p_5=11$  is a good prime since  $11^2=121$  is greater than  $7\cdot13=91$,  $5\cdot17=85$,  $3\cdot19=57$  and  $2\cdot23=46$.

Carl Pomerance has proved that, as Selfrigde conjectured, there are infinite good primes.

The earliest runs of 2, 3,..., 7 consecutive good primes start at 37, 557, 1847, 216703, 6929381, 134193727, 15118087477 (this last value found by Jim Fougeron).

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 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of good primes
Useful links Wikipedia, Good prime
Carl Pomerance,
The prime number graph, Math. Comp. 33 (1979), 399–408.
Good primes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 11 101 + 198461261 198463511 aban 11 17 + 181000663 181000679 alt.fact. 101 alternating 29 41 + 189834343 189834389 amenable 17 29 + 199884017 199968469 apocalyptic 251 541 + 29833 29983 arithmetic 11 17 + 9993317 9993337 balanced p. 53 257 + 199216613 199880953 bemirp 10061 10091 19001 c.decagonal 11 101 + 166897531 167360051 c.heptagonal 71 2647 + 96523883 145857181 c.pentagonal 331 9151 + 87956731 146669851 c.square 41 1861 + 93038441 140868113 c.triangular 4621 1173511 + 157148191 166368739 canyon 101 307 + 98210179 98420579 Carol 223 Chen 11 17 + 99904499 99927257 congruent 29 37 + 9993223 9993317 constructible 17 257 65537 Cunningham 17 37 + 110586257 139240001 Curzon 29 41 + 199882961 199883693 cyclic 11 17 + 9993317 9993337 d-powerful 739 2063 + 9854627 9887573 de Polignac 127 149 + 99902809 99904457 deficient 11 17 + 9993317 9993337 dig.balanced 11 37 + 199771321 199784461 economical 11 17 + 19988347 19988359 emirp 17 37 + 199884017 199968443 equidigital 11 17 + 19988347 19988359 esthetic 67 101 Eulerian 11 65519 evil 17 29 + 199883767 199968443 fibodiv 149 Fibonacci 1597 Friedman 127 347 + 885727 937571 Gilda 29 happy 97 331 + 9990731 9993337 hex 37 127 + 169358047 169989769 Hogben 307 3907 + 143700157 196770757 Honaker 3433 4903 + 198464323 199022107 hungry 17 iban 11 17 + 747377 747401 idoneal 37 inconsummate 431 563 + 996803 999953 Jacobsthal 11 junction 101 307 + 99859147 99904439 katadrome 41 53 + 6521 8521 Kynea 1087 263167 Leyland 17 593 lonely 53 Lucas 11 29 lucky 37 67 + 9990457 9993223 m-pointer 61211 312161 magnanimous 11 29 + 224027 602081 metadrome 17 29 + 13679 1245689 modest 29 59 + 171000481 172001287 Motzkin 127 mountain 191 251 + 124595321 145975421 nialpdrome 11 41 + 98774441 99754411 nude 11 oban 11 17 + 937 967 odious 11 37 + 199968469 199968539 Ormiston 40693 94397 + 199678769 199882217 palindromic 11 101 + 176333671 195353591 palprime 11 101 + 176333671 195353591 pancake 11 29 + 145581517 186331861 panconsummate 11 37 + 257 331 pandigital 11 partition 11 101 pernicious 11 17 + 9993223 9993337 Perrin 17 29 853 Pierpont 17 37 + 3457 65537 plaindrome 11 17 + 125588999 133333777 prime 11 17 + 199968469 199968539 primeval 37 1367 13679 Proth 17 41 + 167723009 190857217 repdigit 11 repunit 127 307 + 143700157 196770757 self 53 97 + 199880953 199883539 self-describing 12103331 sliding 11 29 101 641 Sophie Germain 11 29 + 199882961 199883693 star 37 541 + 66593353 114607621 strobogrammatic 11 101 119611 strong prime 11 17 + 99904573 99927257 super-d 127 269 + 9992681 9993019 tetranacci 29 tribonacci 149 trimorphic 251 truncatable prime 17 29 + 13294397 99537547 twin 11 17 + 199884017 199968539 uban 11 17 + 93000059 93000097 Ulam 11 53 + 9992483 9992681 undulating 101 191 727 929 upside-down 37 9371 + 91864291 98773321 weakly prime 5564453 59160317 61589579 126517543 Wieferich 3511 Woodall 17 191 Zuckerman 11 zygodrome 11 11777 + 2244499 117770011