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Let $ a $ and $ b $ be two positive integers such that $ a\lt b $ and $ ab $ is a primorial. Let $\mathcal{N}(x)=\mathcal{N}_{prime}(x)+\mathcal{N}_{pure}(x)+\mathcal{N}_{mixed}(x)$ where $ \mathcal{N}(x) $ is the number of such pairs $ (a,b) $ with $ b\le x $, $\mathcal{N}_{prime} (x)$ the number of such pairs $ (a,b) $ with $ b\le x$ fulfilling the additional requirement $ (b-a,b+a)\in\mathbb{P}^{2} $, $ \mathcal{N}_{pure}(x) $ defined similarly but with $\Lambda(b-a)\Lambda(a+b)>0 $ and $ \Omega(b^2-a^2)>2 $ and finally $ \mathcal{N}_{mixed}(x) $ the number of such pairs $ (a,b) $ with $ b\le x $ such that $ \Lambda(b-a)\Lambda(a+b)=0 $ where $ \Lambda $ is the Von Mangoldt function and $ \Omega(n) $ the total number of prime factors of $ n $ counted with multiplicities.

Does one have $ \mathcal{N}_{mixed}(x)=o(\mathcal{N}_{prime}(x)) $?

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  • $\begingroup$ Once b gets big, say b larger than 1000, I would expect the primes quantity to be relatively small, especially if you started multiplying a by larger (but not too large) primes. In spite of your definition, I am not clear on what is captured by the other two quantities. You might find generalizing this to coprime and squarefree a and b more tractable, or at least giving a good perspective. Gerhard "Study Weaker To Get Stronger" Paseman, 2018.01.23. $\endgroup$
    – Gerhard Paseman
    Commented Jan 24, 2018 at 0:04
  • $\begingroup$ $ \mathcal{N}_{pure}(x) $ counts the number of $ (a,b) $ such that $ b\le x $, $ b-a=p^m $ , $ a+b=q^n $ for primes $ p $ and $ q $ with at least one of the two positive integers $ m $ and $ n $ being greater or equal than $ 2 $ . The third quantity counts the number of $ (a,b) $ such that at least one of the two integers $ b-a $ and $ a+b $ has at least two distinct prime factors. $\endgroup$
    – Sylvain JULIEN
    Commented Jan 24, 2018 at 10:58

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Certainly $b\pm a$ have no particularly small factors. It would be interesting to know what results you have for moderate size $x.$

Here is some very weak evidence that suggests that, while $\mathcal{N}_{prime}(x)$ might be fairly large compared to what one could naively expect, probably $\mathcal{N}_{prime}(x) \lt \mathcal{N}_{mixed}(x):$

The product of the first 10 primes is $6469693230$ (with square root about $80434$.) This allows $512$ choices for $b \gt a.$ Of these, $106$ have both $b-a$ and $b+a$ prime. Here are the $a$ values:

3, 17, 57, 145, 209, 406, 418, 442, 462, 546, 609, 667, 759, 770, 782, 805, 874, 1015, 1131, 1254, 1330, 1463, 1870, 2233, 2346, 2431, 2717, 2730, 3135, 3230, 3289, 3315, 3451, 3570, 3705, 3770, 4147, 4370, 4485, 4830, 4862, 5278, 5423, 5474, 5681, 5865, 6270, 6279, 6555, 6670, 6783, 7293, 7854, 7917, 10010, 10166, 11339, 13398, 13585, 14007, 14326, 15249, 15470, 15834, 16302, 16422, 17342, 18734, 19227, 19285, 20010, 20930, 23023, 24310, 28014, 28101, 29393, 33649, 34034, 35581, 37145, 37961, 38019, 39585, 40755, 41055, 42427, 43355, 46410, 51870, 54230, 56202, 57057, 58058, 62491, 62985, 64090, 66990, 67298, 67830, 68034, 70035, 74613, 75922, 79373, 79534

ALSO In none of the cases are both $b-a$ and $b+a$ prime powers (except the ones with both being actual primes.)

It is probably no help to you, but I am reminded of the quirky paper Primes at a Glance. In case $ab$ is the product of all primes up to $p$ and $1 \lt b-a \lt p^2$ (or I guess even the square of the next prime) then $b-a$ is clearly prime. That paper shows that such an occurrence is rare and discusses related matters.

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  • $\begingroup$ 106 is more than I would have guessed for that problem. Any examples for the pure category? Gerhard "Needs Prime Sums Intuition Readjusted" Paseman, 2018.01.24. $\endgroup$
    – Gerhard Paseman
    Commented Jan 24, 2018 at 15:19
  • $\begingroup$ Turns out not. I wonder about other small cases. $\endgroup$
    – Aaron Meyerowitz
    Commented Jan 24, 2018 at 15:33
  • $\begingroup$ If one represents the prime powers as c and d, there are some mod 8 obstructions on the pair (c,d) (and maybe interesting obstructions mod m for some m), which may strongly limit the search space for such examples. Gerhard "Power Searching For Prime Powers" Paseman, 2018.01.24. $\endgroup$
    – Gerhard Paseman
    Commented Jan 24, 2018 at 16:04
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    $\begingroup$ For the first nine primes one has that $82339-76729=277^2$ and the sum is prime. No other examples with 10-20 primes. There are three more examples with smaller numbers of primes. $\endgroup$
    – Aaron Meyerowitz
    Commented Jan 24, 2018 at 18:53
  • $\begingroup$ An interesting thing is that $ \log 6469693230 $ is quite close to $ 106/\log 106 $ . $\endgroup$
    – Sylvain JULIEN
    Commented Jan 24, 2018 at 20:54

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