Combinatorial prime puzzle

Question

Is it true that no prime larger than $241$ can be made by either acting or subtracting $2$ coprime numbers made up out of the prime factors $2,3,$ and $5?$

Update

Above example is clearly wrong, as shown by MJD. New question:Is it true that $251$ cannot be made by either acting or subtracting $2$ coprime numbers made up out of the prime factors $2,3,$ and $5?$

@MJD ok - I hold my hands up & say that I certainly deserve the imminent flood of downvotes :/ — martin, Oct 31 '14 at 14:53
@martin Please consider my computer programming suggestion. It takes only a few hours to learn how program the computer well enough to have it check this kind of conjecture, a few minutes to write the program once you know how, and a few seconds to actually run the program. — MJD, Oct 31 '14 at 15:11
@MJD surprisingly, this question was as a result of a mistake in my programming a far more complex problem than need be for this type of question :/ — martin, Oct 31 '14 at 15:18
@MJD I might repoost - since I have just found the source of teh problem. As far as I can see, $251$ cannot be made by adding or subtracting 2 coprimes by using ALL of the prime factors $2,3,5$ ... thought I might just run it by you first though! — martin, Oct 31 '14 at 15:26
@MJD I think it is a case of not seeing the wood for the trees :/ — martin, Oct 31 '14 at 15:47

MJD · Accepted Answer · 2014-10-31T15:01:07.137

4

You mean like $162 + 625 = 787$?

Intuitively, it would very surprising if this conjecture were true. There is no reason to expect that the sum or difference of two arbitrary numbers would not in general be prime, and there are quite a lot of primes, so something surprising would have to happen for this large family of sums and differences to almost completely miss all the primes.

Brute-force computer search finds many counterexamples; for example $2^{19} + 3^4 =524369$. If you are interested in this kind of conjecture, learning a minimal amount of computer programming would be a good investment of your time.

edited Oct 31 '14 at 15:01

answered Oct 31 '14 at 14:42

MJD

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Yes, I posted the question in haste, I am afraid. However, there certainly appears to be a list of primes greater than 241 that cannot be made. – martin Oct 31 '14 at 14:49
Also $1039, 2063, 4111, 32783, 65551$ of the form $2^n+15$ – Mark Bennet Oct 31 '14 at 14:50
2

@martin It's not surprising; the set of numbers that are sums or differences of 5-smooth numbers has asymptotic density 0, so one should expect its intersection with the primes to be 'small'. But there's no reason to expect it to be finite. – Steven Stadnicki Oct 31 '14 at 15:17
1

(A minor correction to my previous comment: the set of sums of 5-smooth numbers has density 0, but I doubt it's known whether the set of differences of 5-smooth numbers does. But I believe that this would follow from the ABC conjecture, so it's fairly plausible. – Steven Stadnicki Oct 31 '14 at 15:27
@StevenStadnicki thanks for the note - I feel slightly embarrased that I didn't check this properly, and I shall take more time in formulating another question in a similar vain! – martin Oct 31 '14 at 15:37
@martin The details on this question are off, but the topic itself is an interesting one and I encourage you to continue looking into it (though I'll warn that you're going to run into a lot of 'nobody knows'!). You might be interested in http://math.stackexchange.com/questions/44163/are-there-infinitely-many-primes-next-to-smooth-numbers as a question in a similar vein... – Steven Stadnicki Oct 31 '14 at 17:50
@StevenStadnicki great - thanks! (vein!!) – martin Oct 31 '14 at 17:53
@StevenStadnicki If you have time, I would be grateful to have your thoughts on the reframed question here – martin Oct 31 '14 at 18:18

Combinatorial prime puzzle

Update

1 Answers1