Dear Dr. Adiya

As far as I know, a Mersenne prime is a prime number that is one less than a power of two.

Thank you Dr. Maan, numbers of the form 2^n-1 are called Mersenne numbers and not necessarily prime, but when 2^n-1 is prime it is said to be a Mersenne prime.

Dear Adiya K. Hussein ,

The best test for Mersenne primes is the Lucas-Lehmer test.

In our new paper in Journal of Number Theory

Volume 182, January 2018, Pages 271-283

"On a new improved unifying closed formula for all Fibonacci-type sequences and some applications"

we have the following theorem which offers a new version of Lucas–

Lehmer test.

Theorem:

Mp = 2p − 1 is prime if and only if cos(2p-2(cos-1 2)) ≡ 0 (mod Mp).

Hope you find it useful.

Best regards

Dear Issam Kaddoura

Thank you very much, I agree with you that Lucas-Lehmer test is the best for Mersenne primes. I am intrested in your paper i find it very useful in my research feild.

There is no formula to generate the next prime number of the form 2^p - 1. If there were, there would be no need to verify that they are prime like what the "Great Internet Mersenne Prime Search" does with distributed computing:

https://www.mersenne.org/

Sorry!

Dear Bruno Martin

This is exact but I'm trying to find a function that enables us to generate

Mersenne prime numbers using a program.

hello Adiya K. Hussein ,

since M(p)=(2^p)-1 it's a good and important problem, and now

the largest known prime number is a Mersenne prime, M(82,589,993), proved prime in 2018.see

https://primes.utm.edu/notes/by_year.html

see also

https://www.johndcook.com/blog/2011/09/09/five-interesting-things-about-mersenne-primes/

best regards.

Dear Adiya K. Hussein ,

An important remark:

If a Mersenne number Mp= 2p - 1 is not prime, then it has a divisor

of the form 2kp+1, k>0.

so if p is prime and Mp is not divisible by any one of the numbers

2p+1, 4p+1, 6p+1,....not exceeding sqrt(Mp),

then Mp is a prime number.

Example:

(1) To test primality of M11 = 2¹¹-1

we test only the divisibility by 2(11)+1, 4(11)+1 ≤ √(2¹¹-1 )

(2¹¹-1)/(23)= 89

therefore, 2¹¹-1 is NOT a Mersenne prime.

(2) To test primality of 2¹³-1

we test only the divisibility by

2(13)+1, 4(13)+1, 6(13)+1 ≤ √(2¹³-1 )

2¹³-1 is not divisible by 27 ,53 and 79,

therefore 2¹³-1 is a Mersenne prime.

Similarly for any Mp.

Best regards

Dear Paul Marcoux

I think the proof is obvious.

Remark: If p is prime and q is a non trivial prime divisor of 2p - 1,

then q = kp +1.

Proof:

Assume that q is a non trivial prime divisor of Mp= 2p - 1,

and p is prime number.Then

Mp ≡ 0 modq ⇒ 2p ≡ 1 modq

Using Little Fermat's theorem, we have 2^{q-1} ≡ 1modq.

Let d=gcd(p,q-1)

If d = p, then q-1≡0 modp and consequently, q = kp+1.

The only other possibility is d =1, p is prime.

Using Bezout's lemma, there exists integers x and y such that

px +( q-1)y = 1.

Consider 2 = 2px+(q-1)y = (2p)x(2q-1)y = (1)x(1)y = 1 modq

this contradiction shows that d = p is the only possible case.

Notice that this property is useful to test small Mersenne primes, but it is not efficient for large exponents p.

Lucas-Lehmer is more efficient than the previous remark, and this can be proved by computing the complexity of each algorithm.

Best regards

Dear Issam Kaddoura ,

It seems possible to formulate a similar criterion for Fermat numbers:

F_n = 2^2^n +1 is prime if and only if cos((2^(2^n-2))(arccos (4))) ≡ 0 (mod F_n).

Sincerely

Dear Predrag Terzić,

Check the notations,

For n =1:

F1 = 5 , but cos(22-2 arccos(4))=cos(arccos(4)) = 4 ≇ 0 mod(5).

Best regards

Dear Issam Kaddoura ,

You are right . I omitted condition: n>1.

Also,

For p=2:

M_2 = 3 , but cos((2^(2-2)) arccos(2))=cos(arccos(2)) = 2 ≇ 0 mod(3).

P.S.

Can you tell me how to produce math expressions like yours? Do you use LaTeX for math rendering or something else?

Dear Predrag Terzić ,

You can use the T2 and T2 notations that appear below.

It helps for better math texts. Best regards

Thank you all for this interesting scientific discussionز

• We are grateful to you, my brother Issam Kaddoura, for all valuable information you have provided about prime numbers.

Dear Peter,

Yes, there are many formulas to generate prime numbers, we have published one of them. ( see the attached paper). BUT, the main challenge is:

No such formula which is efficiently computable is known.

One of the best results was published in

annals.math.princeton.edu/2004/160-2/p12

the authors show that Primes is in P.

Best regards

Mersenne was looking for a formula that would generate all the prime numbers. ... These numbers are now called Mersenne numbers or Mersenne primes. In 1644 he wrote that Mp is prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, and compound – in other words, non-prime – for the other 44 lower p values at 257

In addition, there is a 1-1 correspondence between even perfect numbers

and the Mersenne primes.

That is, if 2p - 1 is Mersenne prime ,then 2p-1 ( 2p - 1) is a perfect number

and the converse is true.

The greatest known Mersenne prime is 2282589933 - 1 discovered in

December - 7 -2018.

Also, It can be seen, that when written in binary form, Mersenne primes consist of only the digit 1. If there are n digits in any Mersenne prime written in binary form, then, the binary number, written as n 1's followed by (n-1) 0's represents a perfect number. as we see here,

Mersenne prime in decimal form, 3, 7, 31, 127, 2047, 8191, 131071, 524287 , ...

Mersenne prime in binary form, 11, 111, 11111, 1111111, 1111111111111, 11111111111111111, ...

Perfect numbers that corresponding to above Mersenne primes,

6, 28, 496, 8128, 335503336, 8589869056, ...

Perfect numbers in binary form,

110, 11100, 111110000, 1111111111111000000000000, 111111111111111110000000000000000, ...

yes, it is a great unsolved problem, and if the even perfect number , with Mersenne prime, the problem How to find perfect number?? it's a good question!, this is a need to answer!

There is a one to one correspondence between Mersenne primes and the even perfect numbers. That is, if 2p - 1 is Mersenne prime, then 2p-1 ( 2p - 1) is an even perfect number.

Conversely, any even perfect number should have the form 2p-1 ( 2p - 1)

where 2p - 1 is a Mersenne prime.

One of the main interesting unsolved problems in number theory:

Is there any odd perfect number?

Here are some interesting properties about the odd perfect number if exist:

(1) It has the form 12k+1 or 36k + 9.

(2) It is not divisible by 3,5 or 7.

(3) It has at least 27 distinct prime factors.

(4) It has at least 101 prime factors.(not necessarily distinct)

(5) It also has the form (4p+1)4k+1 m2 where 4p+1 is prime.

(6) If N is perfect number, then N2-w(N) ≤ (1/3)(2/3)w(N)-1

where w(N) is the number of distinct prime factors of N.

((6) proved by Jose Arnaldo B. Dris, 2012 ).

There are more.

Best regards

Yes,

The new formula(s) generate(s)(calculate(s)) all Mersenne Prime Numbers, and all Mersenne Prime Number Candidates.

Ron Jones

No,

There is no formula to generate all Mersenne Primes.

If so, show your reference!

New Mathematics Conference at

The Johns Hopkins University

April, 24, 2000

Ron Jones, M.D. (Presenter) to (then)

Professor and Chairman John Wierman and

Professor Ed Scheinerman

Department of Mathematical Sciences

Re-presented and Invited Demonstration

Princeton University

July 26, 1997

Ron Jones, M.D. (Presenter) to

John von Neumann Professor in

Applied and Computational Mathematics

Professor John Horton Conway

Ron Jones

You have listed a lot of famous names!

Where is the reference? or any article about generating all Mersenne primes!

To detect Mersenne primes is one of the most difficult open problems!

I think it is not your domain, you may get it wrong.

Very Nice.

Too bad we lost Professor Conway last April-(Covid).

Otherwise, Issam, perhaps you, he, and I might have

arranged to go over some of this together again, at

Princeton, although by now, John would have been

emeritus, and my not have been interested.

I regard your command and skepticism as astute and

appropriate, but some discoveries may not

be reported, especially those with impact in privacy and

cryptology.

Ron

Indeed it is bad news.

John Conway is one of the greatest mathematicians of this century.

He was the inventor of Game of Life.

To his memory, we present the following link for readers:

https://www.newyorker.com/culture/annals-of-inquiry/three-mathematicians-we-lost-in-2020

A True and IncomparableGiant!

Issam are you in or near NYC? If not, when might you be?

Ron

Although, Mersenne primes and Mersenne numbers have huge literature commenting in their properties and uses, however it is equally worthwhile to mention that Mersenne numbers are also not described completely. One may think of perfect numbers associated with Mersenne numbers but that is just one application of mersenne numbers. Questions like why all mersenne numbers are squarefree are still unproven.

Ok, so the set of Mersenne Prime Numbers differs from the set of Mersenne Numbers. Ron

Dear John,

I feel bad about the loss of the great mathematician John Conway.

I agree with you that professor John is (A True and Incomparable Giant!).

Dear Aziz Lokhandwala

Your statement

"Questions like why all mersenne numbers are squarefree are still unproven"

is not correct. You can see the counterexample:

2100 - 1 = 3×5³×11×31×41×101×251×601×1801×4051×8101× 268501

is not square-free.