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Are there infinitely many primes $p, q, r, t$ such as $pq+2=rt$

$3\times{11}+2=5\times{7}$
$5\times{11}+2=3\times{19}$

I guess this may be an open question?

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    It is interesting but this problem is equivalent to: are there infinitely many primes that $ \det{\begin{bmatrix} r & q \\ p & t \end{bmatrix}}=2$. – 2017-01-17
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    One more problem of the cathegory : Very probably true, but very hard to prove. Even if we fix $p=3$, we should have infinite many solutions – 2017-01-17
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    I suppose it's not hard to come up with such expressions but like Peter said, they are hard to prove. – 2017-01-17
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    @Widawensen Then give a survey as an answer to show evidence for the truth of the claim – 2017-01-17
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    Is it required for $p, q, r, t$ to be four distinct primes? – 2017-01-17
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    For what it's worth: neither http://oeis.org/A198327 nor http://oeis.org/A092207 are marked finite. – 2017-01-17

2 Answers 2

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Even the equation

$$3p+2=5q$$ with primes $p$ and $q$ probably has infinite many solutions :

Suppose, $p=5k+1$ and $q=3k+1$ are both prime , then the equation $3p+2=5q$ is satisfied.

If the generalized bunyakovsky conjecture (See https://en.wikipedia.org/wiki/Bunyakovsky_conjecture ) is true there are infinite many such $k$, so the given equation probably has infinite many solutions.

2

The question is whether there are infinitely many integers $n$ such that both $n,n+2$ have exactly two prime factors. Whether this is occurs is an open question.

This is essentially a problem in sieve theory. The parity problem is a fundamental obstruction that prevents us from solving many of these sorts of problems.

Essentially, the parity problem prevents us from distinguishing numbers with an even or odd number of prime factors.