Problems & Puzzles: Puzzles

Puzzle 854. Follow up to Puzzle 853 Emmanuel Vantieghem sent the following nice follow up to Puzzle 853 There might be other polynomials, say  a x^2 +b x +c  with smaller coefficients (for instance, such that  a+b+c  is smallest) that are prime at 0 and 81  and composite in-between. A good starting point is the following solution found by Emmanuel:  x^2 + x + 16064431 is prime at  0  and  81  and composite in-between. So, a+b+c=16064433 Q. Find your a, b, c values for this question such that a+b+c<16064433.

---

Contributions came from Seiji Tomita, Jan van Delden, Shyam Sunder Gupta and Michael Hürter and Emmanuel Vantieghem

***

Seiji wrote:

f=ax^2+bx+c
Search condition: a<100, b<1000, c<10000
N: number of consecutive positive integer
80<=N<100
f(0) and f(N+1): prime number
f(x): composite for 1<=x<=N

[N,  a,  b,  c, a+b+c]
[80, 46, 24, 53, 123]
[81, 91, 76, 3, 170]
[82, 48, 476, 1381, 1905]
[83, 31, 6, 53, 90]
[84, 94, 388, 3, 485]
[85, 71, 124, 3, 198]
[86, 94, 54, 167, 315]
[87, 32, 155, 3, 190]
[88, 87, 997, 6971, 8055]
[89, 29, 24, 7, 60]
[91, 79, 77, 3, 159]
[92, 34, 195, 2, 231]
[93, 96, 145, 61, 302]
[95, 19, 24, 23, 66]
[97, 79, 8, 3, 90]
[98, 98, 78, 109, 285]
[99, 56, 11, 3, 70]

***

Jan wrote:

The polynom 46x^2+24x+53 has minimal

 

Sum:     a+b+c=123

Max:     max(a,b,c)=53

Norm:   a^2+b^2+c^2=5501

***

Shyam wrote:

The values of a,b and c are 46, 24 and 53. so a + b + c = 123 and this is the smallest possible value of a+b+c such that a x^2 + b x + c gives prime at 0 and 81 and composite in-between.

***

Michael wrote:

I have the following solution for Prime Puzzle 854:

46 * x ^2 + 24 * x + 53

a + b + c = 123

***

Emmanuel wrote:

Here is the best I could find about puzzle 854 :
 

The polynomial  77x^2+15x+37
 

is prime for  x = 0  and  81  and composite for  x  in between.

***