Problems & Puzzles:
Puzzles
Puzzle 206. (Nk)/k
primes
Ken Wilke
sent the following puzzle, as a companion piece to
Puzzle 181:
Find the smallest
positive integer N such that (N  k)/k is prime for k = 1, 2, ..., n, for a
given value of n.
Here are the first
Wilke's results:
n 
N 
1 
3 
2 
6 
3 
12 
4 
12 
5 
174600 
6 
7224840 
7 
10780560 
8 
10780560

9 
2360150402400 
10 
50060257410240 
11 
7720634052774720 
Needless to say that
these results are not coming from a totally brute force approach.
Question: Can you extend the above table using your own approach?
Solution:
J. K. Andersen and J. Wiesenbauer found
a smaller solution for n=9 than the reported by Ken Wilke.
Andersen also found solutions for n=12, 13 & 14.
This what Andersen wrote:
I found a single smaller solution than Wilke for
n=9: N=1086338816640 I agree with the rest of his results and add these:
n=12: N=227457297898150320
n=13: N=7272877497848202240
n=14: N=7272877497848202240 (~ 7.27*10^18)
n=15: N>10^21
I wrote a C program to eliminate candidates with
small factors. prptesting (and finally primality proving) of remaining
candidates was done with PrimeForm/GW but the time for this was
insignificant. I prefer to write N/k1. N is obviously divisible by 1, 2,
..., n. For n=12, N is then divisible by 2^3, 3^2, 5, 7, 11, plus 16
because N/81 must be odd. So 16*9*5*7*11=55440 divides N. Let N =A*x,
where A=55440 Then N/k1 = (A/k)*x1 = ak*x1, where a1=A/1, ..., a12=A/12
are constants. A prime p divides N/k1 if ak*x1 == 0 (mod p). The value
of x (mod p) decides this. For all small p, all remainders making p divide
one of N/k1 can be computed and stored during initialization. I used this
for a mix of wheeling, sieving and trial factoring, making optimizations
along the way. Some were only for n=15 without result so far, but I will
let it run some days if nobody else finds it. The speed is around 5*10^19
pr hour but my heuristics say it could still take months. n=14 was lucky
and I hope for more luck. For n=15, I use a big wheel to reduce the
candidates. Only 1261568 x's out of 3706662405 = 3*5*17*19*23*29*31*37
consecutive numbers, or 1/2938, have no N/k1 divisible by one of
3,5,17,19,23,29,31,37. These x's modulo 3706662405 are stored and this
means only one N must be tested for every 2938*(55440*13) ~ 2 billion
integers. These tests could be faster and I may optimize them unless the
solutions show a sieve expert (maybe Phil Carmody?) has already
searched much longer.
I stopped my search for n=15 at N=25*10^21 without finding a solution.
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