Problems & Puzzles: Puzzles

Puzzle 155.  Follow-up of the puzzle 151

In the puzzle 151 we asked for primes in x^2 + x +A for a given A, such that at the same time they are successive regarding the variable x and consecutive as primes.

Now, in this puzzle, we release the condition of consecutiveness of the primes and ask only for:

k primes that are successive in certain range (x1, xk) of the variable x, not necessarily being x1=0 and being A>41.

As a matter of fact Luis Rodríguez is again the promoter of this new question and the author of all the values in the second part of the following table, except the last one entry:

 k A>0 x1=0 Author 1 2 0 Euler 2 3 0 Euler 4 5 0 Euler 10 11 0 Euler 16 17 0 Euler 40 41 0 Euler k A>41 x1=>0 Author 6 107 4 LR 7 251 13 LR 8 101 7 LR 9 437 27 SSG 10 3317 250 LR 11 221 1 SSG 12 15371 63 SSG 13 7517 106 LR 14 5771 3 LR 15 68501 1104 LR 16 1840997 41 SSG 17 170237 5486 LR 18 246405827 41 LR 19 1847381 5775 CR 20 1418661857 19771 LR using a code by J. C. Meyrignac, found the 20/11/01

*     In green color the rows for the known minimal A values for k successive primes
**   Please notice that X1 may be larger than A.
*** For the row k=>15 the maximum X value we have searched is 20,000

Questions

Can you fill and extend the table?

_________
Hints:
1) Please notice that A need not to be necessarily prime, but it must have an ending digit equal to 1 or 7.
2) The Ubasic-code that Rodriguez and myself have been using is available on request by email.

Solution:

Shyam Sunder Gupta has found (13/10/01) several lower A values for certain k, than those reported by Luis Rodríguez:

I have found smaller values of A than what is mentioned in the Table for k=6,7,9,11 and 12 . These are as follows:

k=6, A=101, x1=20
k=7, A=101, x1=436 (I also found k=7, A=377, x1=161)
k=9, A=437, x1=27 (so x1<A)
k=11, A=221, x1=1
k=12, A=15371, x1=63

In particular is remarkable the value for A=221 that changes the "green table" inside the whole table. These values have been added to the table above.

***

J. K. Andersen srote (30/4/03)

Assuming my search program is correct, Rodriguez found the only solution for

k>=20 with primes below 2^32. With an efficient algorithm, it only took

k>an

hour to test all possible combinations of A and x1. I rediscovered Rodriguez' A=1418661857 and x1=19771 but got nothing else. Going above 2^32 requires reprogramming to 64-bit integers. As primes get more rare, I would not expect to beat k=20 in a reasonable time and decided not to try.

***

 Records   |  Conjectures  |  Problems  |  Puzzles