Problems & Puzzles:
Puzzles
Puzzle 155.
Followup 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 Ubasiccode 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 64bit integers. As primes get more rare,
I would not expect to beat k=20 in a reasonable time and decided not to
try.
***
