Problems & Puzzles: Puzzles

Puzzle 136.  Poor prime strings

Here we look for primes P such that A&P (& = concatenation symbol) remains composite for 1<=A<=K while (K+1)&P is prime.

Example: For P = 101 all the following numbers 1101, 2101, 3101 and 4101 are composite but 5101 is prime.

A somewhat larger example is the prime P = 21799079. For this prime A&P remains composite for 1<=A<=127 but 12821799079 is prime

Questions:

1. Find the least prime P such that A&P remains composite for 1<=A<=K, for K= 200, 250, 300?

If P is not a suffix string but a prefix then we have the following example: for P= 9645263, P&A remains composite for 1<=A<=310 but 96545263311 is prime

2. Find the least prime P such that P&A remains composite for 1<=A<=K, for K= 400, 450, 500?

3.- Theoretically how large can be A against P in both cases (P prefix or P suffix)

Solution

Jim Fougeron wrote (12/5/02):

The K=200 answer for 136 question 1 is P=9578630051 Fortunately, the first value at or above K=200 was exactly at K=200. I have pushed the P value up to 40 billion (40000000000), and found these other K's above 200, but none to 250:

A&P=20421616626329 K=203
A&P=22626004456337 K=225
A&P=21333235129159 K=212

For question #2, I have found K=436 for P=1038712373. However, I am not
sure if you are looking for exactly K=400. These were found before
the "exact" K=400:

P&A=1266830797403 K=402
P&A=1926429577417 K=416
P&A=1996427911409 K=408
P&A=2177481401411 K=410
P&A=2320354291433 K=432
P&A=2336687021411 K=410
P&A=2395844161423 K=422
P&A=3050602271413 K=412
P&A=3575121289517 K=516 (Note over both 450 and 500)
P&A=4185986341403 K=402
P&A=4357157921419 K=418
P&A=4605053083403 K=402
P&A=4854350063401 K=400

Also note that we hit the 500 value at 3.5 billion, while for the A&P,
the 200 mark (same "expected" point) was hit at 9.5 billion

For the P&A 450 mark, here was what I found:
P&A=3575121289517 K=516
P&A=5998149907511 K=510
P&A=7977980083481 K=480
P&A=10655978111507 K=506
P&A=10799501431451 K=450

For the K=500, I have found these values searched up to 15 billion:

P&A=3575121289517 K=516
P&A=5998149907511 K=510
P&A=10655978111507 K=506
P&A=11826614069509 K=508

It appears that the P&A are much more likely to be long. One would
expect this. I expect that P&A should be (5/2)x the size of A&P, since
6 out of 10 A's are known to be composite (evens and 5's). In the A&P,
there were never evens, and there were never A&P == 0 mod(5) since 0 mod(5)
implies that P is not a prime (except when P is 5, and for that one example,
there is no non-composite A). So one conjecture about occurrences, is that
P&A is 5/2 longer on average than A&P (but this is not quite right).

However, the P&A appear to be longer than this. The early long results
for the P&A (500 for a P&A should be the same as 200 for a A&P), show
this. Also, the way I have coded my software, I do not test those 5/2
values at all, but skip them. If all were the same, then the rate the
software was running should be about the same. It is not. The rate is
about 680000/s for the A&P and only 480000/s for the P&A run. 95% of
the difference boils down to the number of tests being performed on
average per each prime. I believe this is due to the fact the A&P is
much more of a "random" selection method that P&A. Let me explain:

For P&A, there are 9 sequential, then 90 sequential, then 900 sequential
numbers, ... . For the A&P, there is nothing sequential. You simply
look at some "arbitrary" number which can have no factor of 2 or 5. I
believe the longer average runs for P&A are due to this sequential
behavior of P&A's. Primes have an average distribution, but they also
have a "clustering" effect, where there are long dry spells, with no
sequential primes, and there are clusters of primes at a much higher
density than expected. The P&A shows this behavior, while the A&P hides
it in it's Monte Carlo selection method. The same ideas are behind why
the better ECM factoring software chooses "random" curves to test. A
certain percentage of ECM curves will succeed in factoring a given length
factor, however, these "good" curves are not evenly distributed. They
have "clusters" and "dry spots". A sequential testing of curves can
pay off quickly (if you happen to be lucky enough to start in or near
a good cluster), but sequential may also take a VERY long time to find
a good curve. A "random" curve minimizes this clustering, and the results
are much closer to the "expected" percentage chance for a good curve.
The A&P behaves like the random choosing of a ECM curve.

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