Mr. Islem Ghaffor, a young 22 years old from Algeria, sent
a "formula" -that he claims to have proved- that counts exactly the
quantity of twin primes not greater than a certain given value, 36n^{2}
+60n +21.

Π_{p;p+2}
(36n2 +60n +21) = Gn =The quantity of twin primes not great than 36n^{2}
+60n +21, for n positive integers, 1, 2, ...

More than "a formula" I would say that Islem has produced "an algorithm"
to make such count.

In short, the count for the quantity
of twin primes not larger than 36n^{2}
+60n +21, for a given integer n value, is equal to 6n^{2}
+10n +4 minus the quantity of distinct terms in 4n arithmetic
progressions concisely defined in the Islem's algorithm.

Independently of the practical value
of the Islem's algorithm in order to compute Gn, the most striking
feature is that the developed algorithm **need not to make any
primality test**, just simple arithmetical operations and discard
repeated terms on the 4n AP's involved.

Nowadays Islem only shares with us a paper that shows his formula and three examples
effectively computed, but he prefers not to share in this moment the
claimed 10 pages proof.

Please see directly the mentioned paper
here.

Islem accepts that he has tested
his formula only for n=1, 2, 3 & 4.

This is why Carlos Rivera has made
some preliminary search in order to verify the Islem's formula for
n values larger than 4, and has found that, as a matter of fact, the
Islem's formula gives correct values for n=1 to 18. Due to limitations
of his code Rivera can not check the formula for n values larger than
18.

**Q1. Would you like to make your
own verification, specially beyond the Rivera's limit of search?**

**
Q2. Do you devise any idea about the foundations of this formula? Does
this algorithm hides a kind of Eratosthenes mesh?**

**
Q3. Do you devise a way to prove that Gn computed by the Islem's formula goes to infinite as n goes to
infinite?**

**In short, five different puzzlers
have tested positively this conjecture, up to different extents, and all of
us await to see the formal Islem's proof of it. Nobody has an answer to
Q2 & Q3 as far as December 12, 2014.**

**The paper by Francesca
Balestrieri, 2011, (submitted by Emmanuel's contribution to this
Conjecture) is another example that shows that the twin prime conjecture
may be posed in a different terms, providing the expectative of a
simpler solution for the old twin primes conjecture**

**"In this short
paper we will
show, via elementary arguments, the equivalence of the Twin Prime
Conjecture to a problem which might be simpler to prove"**

**This is the same expectative
produced by the Islem´s approach.**

Q1: I proved Mr. Ghaffor's
conjecture thru n = 10000 (which translates to confirming twin primes
thru 3600600021). Here are the last 3 lines:

Thru 3599160045 (n = 9998), we found: 10857111 vs.
10857111 = 599860008 - 589002897

Thru 3599879997 (n = 9999), we
found: 10859135 vs. 10859135 = 599980000 - 589120865

Thru 3600600021 (n = 10000), we
found: 10861113 vs. 10861113 = 600100004 - 589238891

10861113 is the twin prime
count., calculated by checking all the primes through 3600600021.

600100004 is "G" for n=10000.
589238891 is the exclusion count.

My program took about 13 seconds
to verify the conjecture after computed the primes. (If you build on
the findings of n-1 and only check the new values introduced at the
nth stage, this will make the search vastly more efficient.)

I find it very interesting what Mr. Ghaffor did and that it appears
to work correctly but I don't think it's much more efficient than
the sieve of Eratosthenes. It took be 31 seconds to generate ALL the
primes under that threshold and then iterate through the list to get
the twin prime count..

I wonder if his algorithm can be modified to find prime counts of
prime pairs with other gaps such as n, n+4. A general heuristic
would be very interesting to see.

For Conjecture 72 Q1. I have no idea how to program this in UBASIC so
instead

I made an Excel spreadsheet to do the problem (which I can provide for
verification). The spreadsheet shows the Gn, l(vo;r), 4n AP's,
calculated and actual counts.

On the spreadsheet I went to n=50. Assuming I made no errors, the
algorithm gives correct
values up to n=50.

As far as I can
see, Islem's procedure is correct for n = 1 to 100 and als for
some bigger n. In my opinion, the procedure is right for all n
but I cannot see exactly why.

The method is
very similar to an existing one that is based on the following
theorem ; If 6n-1 and 6n+1 are simultaneously prime, then n is
not of the form 6xy+x-y or 6xy-x-y or 6xy-x-x (with x, y
natural numbers)

Nevertheless,
Islem's numbers (these are the numbers in the given arithmetic
progressions) seem to have no direct relation to these forms.

I would like to
see his proof !