Just use N= 2X + 1 or N = 2X-1 will do the job for odd numbers.

For prime numbers see: https://en.wikipedia.org/wiki/Formula_for_primes

Dear Peter Breuer,

I wonder if there is a formula or a sequence that generates all the composite numbers.

Dear Dr. Muhammad Ali,

I ask if there is a formula or sequence that generates all composite odd numbers and not a formula that generates odd numbers.

Dear Adiya K. Hussein

For any integer m > 0:

If (2m)!+1/(2m+1) ∉ N, then 2m +1, is an odd composite number.

Otherwise, it is a prime number.

One can construct many other formulas based on a given prime formula.

Best regards

Peter Breuer - " I know that there are many formulas for prime numbers" and that formulas can be used to find odd composite numbers --- but searching for a new formula for odd composite numbers leads to obtaining a new method for finding all prime numbers ... it might be better than the known methods.

I agree that if F(n ) generates all odd composite odd numbers, then you can use this formula to construct all odd prime numbers;

It is merely as follows: O - {F(n)}, where O is the set of all odd natural numbers.

You need to show the following:

1.F(n) includes all odd composite numbers.

2. F(n) is not prime for any n.

3. F(n) is not a reconstruction of some known algorithm to generate primes.

Anyway, one can say nothing about the originality of the work in the absence of F(n) explicit formula.

Peter Breuer

pi(c(n)) + n = c(n) -1.

( In fact 1 is neither prime nor composite)

And pi(p(n)) = n.

Thank you very much Issam Kaddoura , This is what I am trying to find.

Thanks Issam Kaddoura for this valuable detail.

Dear Peter Breuer

" I was hoping somebody would mention the product of the 1/(1 - z2/p2) "

The relation between the Riemann Zetta function in the product form and the distribution of primes is still a conjecture. :-)

Best regards

Dear Adiya K. Hussein

F(m,n)= 8m + 8n + 8mn + 4m² + 4n² + 3, m, n ∈ N

includes only composite odd natural numbers.

Can one show that any odd composite number can be expressed in this form?

It is not an easy task. I wish you good luck in finding such a formula.

Best regards

( The claim is that every composite odd number is one less than a square of an even number? Astonishing! ) !!

Yes it is trivial that

8m+8n+8mn+4m²+4n²+3 = (2n+2m+1)(2n+2m+3)

I raised an important question on how to find a similar form that runs over all composite odd primes. Of course, it is not the above form. And I mentioned that it is very difficult to construct such an expression.

It was a contrapositive statement !!

Dear Issam Kaddoura

Allah willing, I am close to having such a form which generates all composite odd numbers.

Dear Adiya K. Hussein

Inshallah, we expect great news. I wish you good luck and novel results. Best regards

Thank you very much Dr. , Allah bless you.

Dear Adiya K. Hussein you can take a look at my publication where I used a formula that generates odd composite numbers.

Method SUBLINEAR SEGMENTED PRIME SIEVE

Dear Marouane Rhafli

Thank you very much, it is a good research.

I am interested in researching on primes and odd composite numbers. I have a number of researches documented since 2013 one them is an easy formula to generate all odd composite numbers.

Best regards

Could you point us to your research please ?

Kind regards,

Please study how God uses Prime and Composite Numbers in the Quran and hopefully you will find a secret path to your desired goal.

Can the answer be found in new classifications of integers based on their number of prime factors?

Example 12 = 2*2*3 so 12 has three prime factors and thus 12 is a 3D Number. Ingeneral:

1 is 0D number.

Primes are 1D numbers.

Semi-primes are 2D numbers.

etc.

PrimeCalculator classifies numbers this and way too with three different options:

Number_a_Index for any prime factors.

Number_a_Index with any prime factors.

Number_d_Index with duplicate prime factors.

Number_u_Index with unique prime factors.

You can download PrimeCalculator from http://qurancode.com

Dear Ali Adams

I would like to direct your attention to the fundamental theorem of arithmetic and the number-theoretic functions such as the functions Ω(n), w(n) that count the number of prime divisors of n. ( With or without multiplicity)

See Prime omega function - Wikipedia

en.wikipedia.org › wiki › prime_omega_function

This may help to improve your propositions.

Best regards

Dear Issam Kaddoura, Thank you very much for that info. I wasn't aware of such functions.

Please see this pattern hidden within numbers of different dimensions but same rank (index) utilizing Fermat's Prime (p = 4n+1 = a^2 + b^2) as well as a new formula if 0 is considered a natural number [Ali's Composite (c = 4n+1 = a^2 - b^2)

http://heliwave.com/NumberDimension.png

Dear Adiya K. Hussein Marouane Rhafli Peter Breuer Muhammad Ali

For anyone who needs to generate composite numbers with unique/duplicate/any N prime factors, you can use http://heliwave.com/Numbers.zip and http://heliwave.com/PrimeCalculator.zip

Also there is a mysterious pattern regarding 91 and reverse numbers:

Quran = 114 chapters = 6236 verses and was revealed over 114 - 91 = 23 lunar years:

23114 ÷ 91 = 254

41132 ÷ 91 = 452

236236 ÷ 91 = 2596

632632 ÷ 91 = 6952

1146236 ÷ 91 = 12596

6326411 ÷ 91 = 69521

See Pattern 91 in http://heliwave.com/114.txt or .doc or .pdf

Please please please read http://heliwave.com.com/114_AR.pdf in Arabic :)

Dear Ali Adams

It seems you have useful information about the lists of Arabic alphabets and their corresponding numerical values.

How many different scales we have? Also, what about Jewish scales to do the same calculations. Have you any idea? Any reference?

Best regards

Dear Issam Kaddoura

I am the developer of a free software called QuranCode downloadable from http://qurancode.com

At one point I had over 920,000 letter-value systems but there should be only 7 correct ones.

The Jewish system Abjad_Gematria was revealed first at the time of Prophet Noah (peace be upon him).

My Primalogy System was revealed in 2008, thank to God forever ....

http://heliwave.com/Primalogy.pdf

http://heliwave.com/Primalogy_AR.pdf

Most of my research centers around finding out "Why each chapter (surah) in the Quran has the number of verses (ayas) it has?" See http://heliwave.com/Symmetry.xls

The latest findings of our group research is always updated at

http://heliwave.com/114.txt

http://heliwave.com/114.pdf

http://heliwave.com/114_AR.pdf

God > infinity

Dear Ali Adams

I appreciate your great efforts on this important topic.

I will revise the documents and share my feedback as soon as possible I can.

Best wishes

Thank you brother Issam Kaddoura

I look forward to your feedback.

God > infinity

Try to attack the problem as Number Dimensions:

1 is 0D

primes are 1D

semi-primes are 2D [e.g. 9=3*3]

tri-prime are 3D [e.g. 12 = 2*2*3]

etc.

http://heliwave.com/NumberDimension.png

Dear Ali Adams

I promised you to read the whole work, and I will do it as soon as I can.

I wish you great achievements.

Dear Issam Kaddoura I love you brother for the sake of Allah

Dear Ali Adams ,

your work is really a great job worth appreciating.

Thank you very much Mrs Adiya K. Hussein for your appreciation.

I wish our religious scholars appreciate it as much. They think I am playing with the Quran (God forbid). Yet, I claim to understand the Quran deeper than their traditional teaching due to my physics and electrical engineering background, al-hamdu liAllah.

Please see http://heliwave.com/Soul.and.Spirit.pdf.

Here is a summary in Arabic: http://heliwave.com/Quran.txt

بسم الله الرحمن الرحيم الحمد كلام والشكر فعل. الله يرزقنا ليختبرنا أنشكر أم نكفر ويوعدنا لئن شكرتم لأزيدنكم. فإشكر الله بمساعدة الـءاخرين. الإنسان بشر ناسىي لعهده مع الله في عالم الذر. الإنسان مذموم في القرءان (جهولا عجولا ظلوما كفورا) وأما البشر الغير ناسي لعهده فلم يهبط الى مستوى إنسان وهو غير مذموم في القرءان. أولوا العزم وأهل البيت ع هم بشر غير ناسين لعهدهم ويوم الحساب يقفون على الأعراف ويعرفون الناس المجمعون ليوم لا ريب فيه كل إنسان بسيماه. النفس هي الأنا وهي التي تُجزى بما كسبت يوم الحساب. الروح محوّل طاقة الى قوة بإذن الله (الذي لا حول ولا قوة إلا به). الروح تحذر النفس عن عواقب ما تنوي من عمل قبل تحويل طاقة الأكل الى طاقة حركية ومد الجسم بها لتنفيذ رغبات النفس. النفس تحيا حينما تنفخ فيها الروح وتموت حينما تسحب منها الروح. الوفاة رجوع تام للنفس الى الله عند الموت أو الشهادة أو النوم. الموت وفاة بسحب الروح ودفن الجسد في قبر ونومه في مرقد بعد 40 يوماً. الشهادة وفاة بدون سحب الروح ومواراة لسَوءة الجسم في ضريح أو بيت أذن الله أن يُرفع ويُذكر فيه إسمه. النوم وفاة يومية بدون سحب الروح وتغطية للجسم في سرير أو مرقد. الشهيد حي ويموت عند نفخة الصور الأولى لأن كل نفس ذائقة الموت. كل الأموات يحيون ليوم الحساب عند نفخة الصور الثانية. جسد الميت يتفسخ بمرور الزمن. جسم النائم ينمو بمرور الزمن. جسم الشهيد جامد في الوقت ولا يتغير مع مرور الزمن. إختبارات الإنسان يختارها الله حسب تصرفاته السابقة المخيرة 100% من البلوغ الى الوفاة الصلاة تطهير للنفس وتقريب الى الله. اللعن طلب لتلويث نفس وإبعاد عن الله. الحلم رؤية مشوهة للنفس الغير طاهرة. رؤى الأنبياء حق لا تشوه. السنة هي سنة هجرية أي قمرية الحول هو سنة ميلادية أي شمسية السنة تعب ومشقة والعام رخاء وسعادة بعث الله نوح ع في قومه وعمره 800 سنة ولبث فيهم ألف سنة إلا خمسين عاما (950 سنة دعوة + 50 عام يصنع السفينة) وعاش بعد الطوفان 700 عام فعمره 2500 سنة القرءان مقرؤء بالرغم من إنه مقفول السورة سور حماية لا تتجزأ ويجب أن تقرأ كاملة في الصلاة الءايات طابوق في السور وبأحجام مختلفة لتعقيد نظام الحماية الفاتحة مفتاح وليس إفتتاح قرءاننا هو القرءان الخارجي العظيم بينما القرءان الداخلي الكريم المكنون لا يمسه (يصله) إلا المطهرون ع بإستخدام سورة الفاتحة وإسم الله الأعظم المكّون من 73 مقطع حيث المقطع الأخير لدى فاطمة الزهراء ع فقط وتوصله الى الإمام المهدي ع عند الرجعة القريبة إن شاء الله الله أكبر من ما لا نهاية ... الحمد لله رب العلمين

God > infinity

http://heliwave.com

Please dear Issam Kaddoura and Adiya K. Hussein download my QuranCode.com software with unlimited number of "scales" or letter-value systems and make sure to click the small + sign) next to the Value field to open the 19 options that may modify each letter/word/verse/chapter values.

See http://heliwave.com/Code19.txt

and http://heliwave.com/Code19.png

Dear Adiya K. Hussein any news about your upcming odd composite generator forumula?

Hi Richard Lanzara your formula (2x-1)^2 + 2(n)(2x-1)  is exactly the one i used in my prime numbers finding algorithm here Method SUBLINEAR SEGMENTED PRIME SIEVE

Richard Lanzara thank you

Zobia Jabeen

Your mentioned article includes an elementary procedure to detect prime numbers. It is well-known before 2000 years called Sieve Of Eratotheness.

The mentioned procedure is inefficient for large numbers.

Regards

Zobia Jabeen

Nobody cares about composite numbers, either odd or even.

Due to the fundamental theorem of arithmetic, any positive integer can be expressed uniquely as a product of prime numbers. (up to order).

So, the main issue is to study the prime distribution. The expression of other integer forms is a trivial consequence of this theorem.

Ex: N = (2k+1)(2m+1)....(2r+1) is always a compositite odd number

for k,m,..r are positive integers.

Similarly,

N=2mnp....r is an even composite integer.

One can see that it is not difficult to construct formulas for odd and even composite numbers. Can you determine an explicit formula for primes?

Here is the main issue!

Zobia Jabeen I read your article, it is not more than the Sieve of Eratosthenes as said by Issam Kaddoura , you didn't find a formula, it is just an ALGORITHM, which requires N numbers of iterations, an improvement to this sieve has already been published in this article Method SUBLINEAR SEGMENTED PRIME SIEVE

If you would like to come up with something new, I invite you to read first what has already been discovered and the improvement made to the famous sieves.

Good luck

I have posted this, I believe I found the equations that generate all composite numbers.

The 5 equations that generate all composite numbers

https://www.vixra.org/pdf/2207.0027v2.pdf

I have this F(n) , but first, i want to publish my work then i will show it.I am looking for a suitable journal so that i may be able to publish my work. This work is really fantastic work.After this ,there is no need to more focus on odd composite , just focus will be on prime numbers.I have done all really, i am just waiting for publishing

https://www.academia.edu/91759870/AIA_Sieve_For_Prime_Numbers_By_Creating_General_Formula_for_Odd_Composite_Numbers_and_Definition_of_Odd_Composites

Following my earlier comment, I have published a paper on the topic of calculation of composite numbers

https://ijmcr.in/index.php/ijmcr/article/view/533/444

One Equation ! My Discovery

Abstract: This paper presents one equation that generate all existing odd composite numbers and this equation can test a number whether it’s a prime or not with 50% less calculation than all previous methods. This paper's focus on Prime numbers has always remained a matter of fascination to the mathematicians, and many scientific and technical communities The idea behind this paper is to create a method for finding prime numbers, as all prime numbers are odd numbers (except for number 2) so finding a formula to generate all odd composite numbers was important. In this paper we introduced a new single equation called “AIA-Equation” that generate all odd composite numbers. Using this equation, composite numbers of any range can be calculated without any restriction. A specific observation is measured and recognize a pattern and generalize it for numbers. We have used AIA-Equation to generate large composite numbers of different range also. The Fundamental purpose of this research paper is to generate all composite numbers by simplest single equation without any limitation.