Problems & Puzzles:
Puzzle 853. The Euler
For sure you already know that the Euler
polynomial f(x)=x^2+x+41 has the striking feature that produces
exactly 40 distinct primes for the following 40
consecutive values for the variable x: 0<=0<=39.
Moreover, the same polynomial produces again the
same set of primes for the range: -40<=x<=-1. Accordingly, x^2+x+41
produces exactly 80 prime numbers (only 40 distinct) for -40<=x<=39.
Hans Rosenberg -in his splendid book "Mathematical
Gems -II" - provides a method (*) that we may use to produce
at least 80 composite f(x) values, for 80 consecutive values
But his method has two -let's say- back draws:
a) The x values are very large!
b) The composites f(x) are at least, not exactly 80;
that is to say, at both extremes of the range of these 80 x values,
there are some more composite values for f(x) for contiguous x
values to given range.
So we ask here a different question;
Q. Find the smallest
range of 80 consecutive x positive integer values that produces
exactly 80 composite values of f(x).
(*) See the solution given in
pp 163-164 to Question 3 posed in page 37
Contributions came from Emmanuel Vantieghem, Shyam Sunder Gupta, Jan van
Delden, J. K. Andersen, Dmitry Kamenetsky and T. D. Noe.
All of them found the earliest asked range:
[187162265,187162344] is mapped on 80 composites by f,
& f(187162345) are primes.
Each of them, added several issues:
1) There are other polynomials
that have smaller solutions.
The best one is x^2 + x
+ 16064431 which is prime at 0 and 81 and composite
2) In addition to above some of the other values
of x such that the range x to x+79 produces exactly 80 composite
values of f(x) are:
3) The first case of at least 80 is 81 starting
at 136209970. (J. K. Andersen )
382218109, 477924204, 565703010, 699761854,
844947792, 854383032, 959576854, 1063251333, 1108580001,
1200522911, 1214182438, 1447123019, 1449252947, 1454176497,
1586126582, 1643146826, 1656292044, 1697085288,
1842492529, 1859047917, 1864386863,
1876808094, 1894537403, 1979963275... (Shyam)
4) The next sets start with x=382218109,
477924204, 699761854, 844947792.(Dmitry)