Thread: Additions to primes that always yield composits

I have noticed that when adding positive integers to prime numbers, the result may or may not yield another prime number. There does not seem to be a pattern as to when an addition yields another prime. However, there is a set of numbers that does not appear to ever yield another prime when added to a prime.

I do not know how large this set is, and have so far not found any particular pattern within this set, but for numbers from 1 to 360 there are 109 numbers that, when added to any prime, do not yield another prime. I have verified this for the first 500,000 primes.

From this set, 50 are primes themselves and 59 are not.

Also, when mapped on a circle, 80% of these 109 numbers appear 180 degrees opposite of each other.

Has anyone else ever played around with this?

- WV

Originally Posted by willemv

I have noticed that when adding positive integers to prime numbers, the result may or may not yield another prime number. There does not seem to be a pattern as to when an addition yields another prime. However, there is a set of numbers that does not appear to ever yield another prime when added to a prime.

I do not know how large this set is, and have so far not found any particular pattern within this set, but for numbers from 1 to 360 there are 109 numbers that, when added to any prime, do not yield another prime. I have verified this for the first 500,000 primes.

From this set, 50 are primes themselves and 59 are not.

Also, when mapped on a circle, 80% of these 109 numbers appear 180 degrees opposite of each other.

Has anyone else ever played around with this?

- WV

Prime gaps see this and this.

RonL

Thanks for the response, but I'm not referring to the gaps between primes or their size. I'll try to explain it more abstractly.

Premise: Any positive integer (prime or not) added to a prime may or may not yield another prime.

P1 + i (= | !=) P2

Observation 1: Positive integers added to a particular prime that yield another prime will not necessarily do so when added to other primes.

if P1 + i = P2 then P2 + i (= | !=) P3...

Observation 2: There are particular positive integers that, when added to a prime (verified for the first 500,000 primes), never yield another prime.

(P1...Pn) + (i1...in) != P

Observation 3: The integers of observation 2 can be prime or composite (a 50/50 ratio may be likely) and therefore do not relate to prime gaps.

So while there is no set of fixed positive integers that when added to any prime will always yield another prime, there appears to be a set of fixed positive integers that when added to any prime will never yield another prime.

I am not able to help you, but I am curious to see an example? Thanks.

-Dan

As mentioned initially, I have only worked with positive integers from 1 to 360. This set yields a subset of 109 positive integers that do not produce a prime from the sum (P + i) for any of the first 500,000 primes. The remaining 251 will produce a prime from this sum at least once.

Creating a grid with 1...360 horizontally and the primes vertically, showing a black dot where the sum produces another prime, results in an intriguing pattern, indicating that some numbers i yield a prime from the sum (P + i) at a much higher percentage than others.

Of course, I would never rule out that I have made a mistake somewhere.

I'd love to dig deeper to see if there is a reoccurring pattern, but checking 1...360 for the first 100,000 primes already takes 7 minutes. If anyone has a very fast primality checker written in PHP that they'd like to share...

The subset of 109 positive integers:

7 13 19 23 25 31 33 37 43 47 49 53 55 61 63 67 73 75 79 83 85 89 91 93 97 103 109 113 115 117 119 121 123 127 131 133 139 141 143 145 151 153 157 159 163 167 169 173 175 181 183 185 187 193 199 201 203 205 207 211 213 215 217 219 223 229 233 235 241 243 245 247 251 253 257 259 263 265 271 273 277 283 285 287 289 293 295 297 299 301 303 307 313 317 319 321 323 325 327 331 333 337 339 341 343 349 353 355 359

BTW, this set of 109 has its own gaps, which so far are always 2, 4, or 6, and does not follow the prime gap pattern.

Originally Posted by willemv

As mentioned initially, I have only worked with positive integers from 1 to 360. This set yields a subset of 109 positive integers that do not produce a prime from the sum (P + i) for any of the first 500,000 primes. The remaining 251 will produce a prime from this sum at least once.

Creating a grid with 1...360 horizontally and the primes vertically, showing a black dot where the sum produces another prime, results in an intriguing pattern, indicating that some numbers i yield a prime from the sum (P + i) at a much higher percentage than others.

Of course, I would never rule out that I have made a mistake somewhere.

I'd love to dig deeper to see if there is a reoccurring pattern, but checking 1...360 for the first 100,000 primes already takes 7 minutes. If anyone has a very fast primality checker written in PHP that they'd like to share...

The subset of 109 positive integers:

7 13 19 23 25 31 33 37 43 47 49 53 55 61 63 67 73 75 79 83 85 89 91 93 97 103 109 113 115 117 119 121 123 127 131 133 139 141 143 145 151 153 157 159 163 167 169 173 175 181 183 185 187 193 199 201 203 205 207 211 213 215 217 219 223 229 233 235 241 243 245 247 251 253 257 259 263 265 271 273 277 283 285 287 289 293 295 297 299 301 303 307 313 317 319 321 323 325 327 331 333 337 339 341 343 349 353 355 359

Ah! I see. The trick is to let the "composite maker" be odd, then the sum of this number plus any odd prime is even, which is composite. So the only number to check is to see if 2 plus the composite maker is composite. These ought to be easy to check out then.

There are an infinite number of primes and (I believe) it has been shown that the frequency of prime twins (prime numbers that have a difference of 2, such as 11 and 13) decreases as the numbers get higher. So it seems fairly reasonable to me (I'm not going to prove it, but if my fact about the twin primes is right, then it shouldn't be too much trouble to make one) that you can make a list of odd numbers and subtract two from each of them. If the answer is a prime, then throw that number out from the list. In this way you should be able to generate an infinite number of composite makers.

-Dan

"you can make a list of odd numbers and subtract two from each of them. If the answer is a prime, then throw that number out from the list"

Hmmm... I do not understand the subtraction of 2 before primality checking, rather than checking for primality immediately. All you do is shift to the previous odd number. Either way you end up with the same result: a list of odd, primeless, positive integers.

I'm not clear on how this directly relates to my observations, as there are both primes and composites that fullfil (P1...Pn) + (i1...in) != P.

Adding 1, 3, 5, or 11 to a prime number will yield another prime at least once for the first 500,000 primes. Yet, 7 and 13 won't. Likewise, 15, 17 and 23 will, while 19, 23, and 25 won't. In the list you propose, 7, 13, 19, and 25 would have been discarded, while 17 and 23 would have been maintained.

"Composite makers" are necessarily odd, but not all odd prime numbers i or all odd composite numbers i fullfil (P1...Pn) + (i1...in) != P.

Perhaps I misunderstood you?