[Math] Pattern in twin primes

linear algebraprime numbersproof-writingsolution-verification

I recently noticed a pattern in twin primes. My questions are: does this pattern continue to hold indefinitely, and how would I prove it? Here's the pattern:

For the $n$th prime, there exists exactly $n-2$ twin prime pairs that can be created as follows:

$p_n$ is the $n$th prime,

$P_p=\prod_{1<m<n}p_m$

$(p_nP_p-4, p_nP_p-2)$

Here's what I've worked up to:

$n=3$ $(p_n=5)$ has $3-2=1$ twin prime pairs
$P_p=3$ gives $(15-4,15-2)=(11,13)$

$n=4$ has 2
$P_p=3$ gives $(17,19)$
$P_p=3\cdot 5$ gives $(101,103)$

$n=5$ has 3
(29,31), (227,229), (1151,1153)

$n=6$ has 4
(191,193), (269,271), (2141,2143), (2999,3001)

$n=7$ has 5
(659,661), (2801,2803), (4637,4639), (23201,23203), (255251,255253)

Again, my questions are: does this pattern continue to hold indefinitely, and how would I prove it?

P.S. How about a pic of a brute force Mathematica script up to like 20?

Best Answer

This is an answer to your last demand:

Do[
 q = Prime[n] Times@@@Rest[Subsets[Table[Prime[k], {k, 2, n - 1}]]];
 twin = Intersection[Select[q - 4, PrimeQ] + 2, Select[q - 2, PrimeQ]];
 Print["n = ", n, " - ", twin, " - ", Length[twin]],
 {n, 3, 20}]

The pattern breaks at $n=9$, since there are $8$ and not $7$ pairs of twin primes. For $n=20$ there are $2674$ pairs.

I have included the possibility of $m=2$, since you use it (although in the question you say $2<m<n$.)

Related Solutions

[Math] Disprove the Twin Prime Conjecture for Exotic Primes

The standard conjectures imply that there are infinitely many primes of the form $p=n^4+2$. No such prime can be part of a twin pair, since $p-2=n^4$ and $p+2=n^4+4=n^4+4n^2+4-4n^2=(n^2+2)^2-(2n)^2=(n^2+2n+2)(n^2-2n+2)$.

A simpler example is $p=n^2-2$.

An even simpler example is $p=21n+5$, and here we don't need any conjectures --- we know that there are infinitely many such primes $p$. Many more examples can be constructed along the same lines, e.g., $15n+7$, $15n+8$, $77n+9$, $39n+11$, etc., etc.

EDIT: The above was written before OP edited the question to indicate interest only in the five types of prime at the top of the question. So, let's look at those. In all cases, please ignore tiny counterexamples to generally true statements.

Mersenne primes $q=2^p-1$, $p$ prime. Trivially $q$ can't be the smaller of a pair of twin primes, so we are asking about $2^p-3$ and $2^p-1$ both being prime. Apparently this does happen from time to time, so there's no simple reason why it shouldn't happen infinitely often. On the other hand, we don't even know there are infinitely many Mersenne primes, so we're not going to prove it does happen infinitely often. In short: hopeless.

Sophie Germain primes: $p$ such that $p$ and $2p+1$ are both prime. $p-2$ is a multiple of 3, so we are asking whether there are infinitely many $p$ such that $p$, $p+2$, and $2p+1$ are all prime. The standard conjectures (e.g., Schinzel's Hypothesis H) say yes, but no one has a clue as to how to prove this. In short: hopeless.

Fermat primes. Maybe there are some congruences to show $2^{2^{2n}}+3$ can't be prime for sufficiently large $n$. It's worth a look. On the other hand, maybe there are only 5 Fermat primes anyway. There are heuristic arguments suggesting that there are only finitely many.

Regular primes. Another set that hasn't been proved infinite, although the smart money is leaning that way. I can't imagine any relation between the regularity of $p$ and the primality of $p\pm2$. Possibly just ignorance on my part, but I'm going to call this one: hopeless.

Fibonacci primes. The Fibonacci numbers grow exponentially, just like the powers of 2 (only not quite as fast), so the situation here is comparable to that with the Mersenne numbers. Hopeless.

[Math] Possible method to prove infinite twin prime conjecture

Wolfram Language code to test whether does $p$ provide the twin prime number pair via your conjecture or not is as follows.

twinPrimesQ[tp_]:=tp[[1]]+2==tp[[2]]&&PrimeQ[tp[[1]]]&&PrimeQ[tp[[2]]];
primesList[p_]:=Module[{out={Prime[3]},i},
    For[i=4,Prime[i]<=p,i=i+1,
        out=Append[out,Prime[i]];
    ];
    out
];
testPrime[p_,pl_]:=Module[{out,found=False,twinPrimes,primeFactors,primeFactorsPowers,i},
    twinPrimes={
        {3 5 prod p-4,3 5 prod p-2},
        {3 5 prod p+2,3 5 prod p+4}
    };
    primeFactors=primesList[p];
    primeFactorsPowers=Tuples[Range[0,pl],primeFactors//Length];
    For[i=1,i<=Length[primeFactorsPowers],i=i+1,
        out=twinPrimes/.prod->Product[primeFactors[[k]]^primeFactorsPowers[[i]][[k]],{k,1,primeFactors//Length}];
        found=twinPrimesQ[out[[1]]]||twinPrimesQ[out[[2]]];
        If[found,Break[]];
    ];
    If[found,out~Select~(twinPrimesQ[#]&)//First,False]
];

This defines a function twinPrime[p,pl] where p=$\,p$ and pl is the maximum power of prime factors of $P_p$ to search upon. The function returns the first found twin pair or False if it has failed.

For example:

You can try this online with Mathics.

To confirm or disprove your conjecture for ranges of primes, you can use

out=List[]; For[i=4,i<=7,i=i+1,out=out~Append~{i,Prime[i],testPrime[Prime[i],1]}]; out//TableForm

adjusting the bounds of search (values of j in terms of consecutive number of primes) and maximum power of prime factors. This will output a table with three columns: the id of the prime being tested, the prime itself and the first twin prime pair found (or False if none).

Best Answer

Related Solutions

[Math] Disprove the Twin Prime Conjecture for Exotic Primes

[Math] Possible method to prove infinite twin prime conjecture

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