618 Strings of prime numbers
Appendix: Long chains of primes
Beiler [p.220] also considers the cubic functionf(n)=n^3 +n^2 +17,
and noted that forn=− 14 ,− 13 ,···, 10, the string of 25 values are all
primes. This is true only when we take− 1 as a prime, sincef(−3) =
− 1. Even if we break the string into two, we still get two long chains of
primes:
f(−14),f(−13), ...,f(−4)a chain of 11 primes.
f(−2),f(−1),f(0), ...,f(10). Butf(0) =f(−1) = 17, we only
have 12 distinct primes.
Beyond these, the longest strings of primes have 6 members. The first
of these begin withn= 717.
Note that on the negative side, there is a string of 10 consecutive
primes from− 183 to− 174. Replacingnby−nwe considern^3 −n− 17
forn= 174, ..., 183 :
nn^3 +n^2 + 17 factorization
173 5147771 683 × 7537
174 5237731 prime
175 5328733 prime
176 5420783 prime
177 5513887 prime
178 5608051 prime
179 5703281 prime
180 5799583 prime
181 5896963 prime
182 5995427 prime
183 6094981 prime
184 6195631 13 × 476587
185 6297383 primeHiggins: 40 primes fromg(x)=9x^2 − 231 x+ 1523,x=0,..., 39.
orh(x)=9x^2 − 471 x+ 6203give the same primes in reverse order.