Problems & Puzzles: Puzzles


Puzzle 853. The Euler polynomial again

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 of x.

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,
and  f(1871622654) & 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 between. (Emmanuel)

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:
382218109, 477924204, 565703010, 699761854, 844947792, 854383032, 959576854, 1063251333, 1108580001, 1200522911, 1214182438, 1447123019, 1449252947, 1454176497, 1477164091,
1586126582, 1643146826, 1656292044, 1697085288, 1842492529, 1859047917, 1864386863,
1876808094, 1894537403, 1979963275... (Shyam)

3) The first case of at least 80 is 81 starting at 136209970. (J. K. Andersen )

4) The next sets start with x=382218109, 477924204,  699761854, 844947792.(Dmitry)




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