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I am looking for theorems or conjectures that may exist regarding adding prime numbers. In particular, about "adding two or more primes, plus a fixed value, that results in a prime number". I am observing an interesting pattern with a set of numbers I am working with and would like to confirm this is a well known "phenomenon".

However, Googling didn't yield much on this specific topic.

I would appreciate any references.

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Notes: Thanks for your commentary. Here are some links on the topic:

Goldbach's conjecture
The On-line Encyclopedia of Integer Sequences - Primes
The On-line Encyclopedia of Integer Sequences - Sum of two primes

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Latest edit:

Link to a related question with full explanation of the observable patter of primes and semiprimes...

Aleksandar M
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edaus
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    "In particular, about "adding two or more primes, plus a fixed value, that results in a prime number". -- Wouldn't that amount to some formula that (easily) generates primes and is thus is well known to probably not exist? "I am observing an interesting pattern with a set of numbers I am working with" -- To better help you, wouldn't it be more prudent to ask about the pattern in question and display what you've observed? – PrincessEev Jul 26 '20 at 08:20
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    We know no way to efficiently generate new huge prime numbers. To find a new prime, we can only construct or randomly (in some range) choose a number without a small factor. If we are lucky, it turns out to be prime. For small numbers , many patterns can be recognized , since many small numbers are prime. But for larger numbers, probably all those patterns will disappear. – Peter Jul 26 '20 at 08:29
  • Thanks for your comments. Yes, I would like to find out if such a formula exists. Even if only for small numbers. The pattern is simply prime1 + prime2 +/-1 = another prime; for primes meeting some criteria. – edaus Jul 26 '20 at 08:40
  • Of course , by choosing the two primes such that they have given residues modulo $2,3,5,\cdots$ upto some prime $p$, we can guarantee , with a suitable choice of the residues , that the expression will have no prime factor upto $p$ , which gives high chances for the expression to be prime. But still, the numbers we get need not be prime. – Peter Jul 26 '20 at 08:55
  • Thanks Peter. So, you are saying, that this is such a basic approach and well known "cluster of patterns" that it doesn't warrant 'a special mention' on its own in prime number theory? It will save me from going deeper into this rabbit hole :-) – edaus Jul 26 '20 at 09:06
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    @Peter . An extreme example: 7 of the first 10 odd numbers are prime so I conjecture that most odd numbers are prime. :). – DanielWainfleet Jul 26 '20 at 09:11
  • Point taken. Yes, pattern breaks for primes lower than 11... and at least one more exception but there could be more exceptions as numbers are going very high. I was just hoping someone has already gone into this rabbit hole so I don't have to. :-) – edaus Jul 26 '20 at 09:18
  • @edaus This rabbit hole (to speak with your words) is unlikely to have an exit. Of course , you can still search for patterns for small numbers , it is hard to determine whether someone else has done this as well (chances are the higher the smaller the numbers are). For computational purposes, this has not a big merit since there are primality tests that need almost no time for numbers upto, lets say , $10^{20}$ and are still very fast for numbers with several hundred digits. – Peter Jul 26 '20 at 09:36
  • @Peter That's partly why I asked this question - I am not a mathematician and I don't even know where to start looking if someone has done this work already or not. To see if this has any merit, the easiest would be to try adding the two largest known primes +/-1 and checking if it is a prime. While adding is trivial, checking if it is a prime is not :-) – edaus Jul 26 '20 at 09:49
  • Yes, I can only agree. Finding so large primes requires a fast computer, much patience and much luck. In principal, everyone can find a record prime, but you should rather search for interesting smaller primes. The idea of adding primes and adding / subtracting $1$ is at least a bit unusual, so chances to find something new are not bad. Whether such primes will be "interesting", is something else. For a start, learn to use pari/gp , a special number theory software , it is easy to program it and to find small primes. – Peter Jul 26 '20 at 09:57
  • You can save much time by starting with trial division, imagine you check a huge number several weeks and then it turns out that it is divisible by $109$ :) This way , you can rule out quite a number of numbers, the rest are the "candidates". – Peter Jul 26 '20 at 10:03
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    It is widely believed, but not proved, that every even number exceeding two can be written as a sum of two primes. It follows from this conjecture of Goldbach that every prime exceeding three can be written as the sum of two primes, plus one, and every odd prime can be written as a sum of two primes, less one. – Gerry Myerson Jul 26 '20 at 10:33
  • @Peter Thanks for additional comments and software recommendation. I will explore it more. – edaus Jul 26 '20 at 11:12
  • @Gerry So, from Goldbach conjecture, "every prime CAN BE the sum of 2 primes +1"? I can definitely see this pattern on small numbers. But not EVERY sum of 2 primes +/-1 will give a prime. For example adding 113 + 131 +/-1 not equal prime (there are more numbers like this and this is the exception in pattern I mentioned earlier). – edaus Jul 26 '20 at 11:23
  • Every even number is (assuming Goldbach) a sum of two primes, but not every even number is within one of a prime, so obviously not every sum of two primes is within one of a prime. A smaller example is $19+7\pm1$, neither choice of sign gives a prime. But to know which sums of two primes are within one of a prime, if that's what you want to know, is (I should think) going to be as hard as knowing which numbers are primes, that is to say, very hard, indeed. Continued... – Gerry Myerson Jul 26 '20 at 12:46
  • ...continued. "I am observing an interesting pattern with a set of numbers I am working with...." If you aren't going to tell us about this "interesting pattern", we really can't tell you anything useful about your problem. – Gerry Myerson Jul 26 '20 at 12:47
  • You are ignoring me. What "interesting pattern" have you observed? – Gerry Myerson Jul 28 '20 at 09:16
  • Sorry @Gerry, not ignoring your question, I was just cut off the internet for a while... "The pattern" here refers to that addition of primes +/-1 and, which of those result in a new prime and which don't. Hence my first line of enquiry was "what is already well known about this phenomenon"... Not to complicate it too much for now, it all stems from a strangely symmetrical "geometric distribution" of numbers which are prime candidates. I don't know the implication of all of this as yet, and if indeed there is "something to it" or not. I am at a very first step of this inquiry... – edaus Aug 07 '20 at 05:28
  • "I am not a mathematician and I don't even know where to start looking if someone has done this work already or not." We could probably tell you whether someone has done this work already IF YOU WOULD ONLY TELL US WHAT WORK YOU ARE DOING. Please, come back when you are prepared to be a little more forthcoming about whatever it is that you are doing. – Gerry Myerson Aug 07 '20 at 06:13
  • Fair enough @Gerry… I can only say that I came across an interesting pattern of prime number candidates by accident and now I am intrigued to research it more and find out ‘what is it good for’. In particular, by arranging numbers in a certain way, I got repetitive blocks of numbers where prime p (or multiples of primes: px times py times… pn) ‘sit’ in only a subset of spots in that block. And this pattern goes into infinity. So unlike with Ulam spiral or other observable patterns, there is ‘order’ that can be easily computed for any block of numbers, no matter how large.... – edaus Aug 08 '20 at 03:01
  • cont... And where there is ‘order’ there is a chance to discover ‘easier way’ of generating primes, checking if a number is a prime, or factorizing large numbers into component prime numbers… My original question refers to one aspect of application of that pattern: that is, I can use it to easily tell if sum of two primes +/- 1 is a prime number candidate (ie. either a prime or multiplication of primes) or other composite number – so a large proportion of numbers can be easily eliminated from further checks. – edaus Aug 08 '20 at 03:01
  • Show us what you've done, edaus, or it's all just a lot of hot air. – Gerry Myerson Aug 08 '20 at 12:06
  • So, @Gerry, if I would like to formalise my conjecture about the distribution of prime numbers and their multiplies (ie. "publish" it), what can you recommend, how should I go about it? – edaus Aug 10 '20 at 01:48
  • Just write it up, coherently, and edit it into the question you have posted here. – Gerry Myerson Aug 10 '20 at 07:55
  • So, it is just hot air, isn't it? – Gerry Myerson Aug 11 '20 at 11:12
  • Hi @Gerry. No, just delayed. I am in correspondence with a local academic specialising in "all things primes". Meantime, also doing some more head scratching regarding potential applications of my conjecture. I will keep you in the loop. – edaus Aug 13 '20 at 02:08
  • Then I don't think this question serves any useful purpose here. Why not delete it, and come back when you have something substantial to say? – Gerry Myerson Aug 14 '20 at 07:24
  • Hi @Gerry, my question is straightforward and still not answered. ;-) Subsequent discussion is on much wider range of issues. I will keep it open for now and "ping" you when I post specifically regarding my own conjecture. – edaus Aug 15 '20 at 06:24
  • Hi @Gerry, it took a while but I finally posted the details: https://math.stackexchange.com/questions/3816005/orderly-distribution-of-primes-and-semiprimes-trivia-or-of-some-practical-use – edaus Sep 06 '20 at 08:28

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