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.

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All of them found the earliest asked range:

[187162265,187162344]  is mapped on  80  composites by  f,
and  f(1871622654) & f(187162345)  are primes.

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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 )