624 Strings of composites
23.3 Consecutive composite values ofx^2 +x+41 .....
Problem 142 ofCrux Mathematicorumasks for 40 consecutive positive
integer values ofxfor whichf(x)=x^2 +x+41are all composites.
Several solutions were published. Unfortunately these numbers were
quite large, being constructed by a method similar to the one above. For
example, here is one. Since forn=0,..., 39,f(n)≤f(39) = 1601,if
we setxn= 1601! +n, then
f(xn)=x^2 n+xn+ 41 = (1601!)·(1601! + 2b+1)+f(n)is a multiple off(n)which is greater thanf(n). These numbers are
therefore composite. These numbers are quite large since 1601! has
4437 digits. TheCruxeditor wrote that “[i]t would be interesting if
some computer nut were to make a search and discover the smallest set
of 40 consecutive integersxfor whichf(x)is composite”.
A near miss is 176955. The string of 38 consecutive numbers begin-
ning with this all give compositef(x). H. L. Nelson, then (and now) ed-
itor ofJournal of Recreational Mathematics, found this smallest string,
with factorization of the correspondingf(x). It begins with 1081296.
There are longer strings.^2 Up to 5,000,000, the longest string of
composites has 50 numbers. There are three such strings, beginning
with 2561526, 3033715, and 3100535 respectively. See Appendix 2 for
the first of these strings.
How about long strings of primes? They are relatively few. The only
strings of≥ 10 consecutive primes begin with 66, 191, 219, 534, and
179856, and no more up to 5,000,000. Each of these strings contains 10
primes, except the one beginning with 219, which contains 13 primes.
(^2) For example, beginning with 1204431, we have a string of 45 composites.