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May 9

Can all integers >1 coprime to 6 be written as sum of a 3-smooth number and a prime?

Prove or disprove: All integers >1 coprime to 6 can be written as sum of a 3-smooth number and a prime. 218.187.64.147 (talk) 13:38, 9 May 2023 (UTC)[reply]

Try getting some estimate for the number of 3-smooth numbers less than N and the number of primes less than N.The estimates don't have to be good. The product will give the number of sums of one and the other. Compare that to the number of numbers less than N which are coprime to 6 NadVolum (talk) 16:53, 9 May 2023 (UTC)[reply]

Does this argument take into account that a given number may be expressed as such a sum in multiple ways? For example, 115885 = 2 + 115883 = 6 + 115879 = 8 + 115877 = 12 + 115873 = ... = 104976 + 10909, altogether 50 different ways. --Lambiam 20:27, 9 May 2023 (UTC)[reply]

No sorry it doesn't - it just gives an upper bound. But that's quite good enough for this. It's fine and easier to use a generous overestimate for the number of 3-smooth numbers below a limit too. NadVolum (talk) 21:55, 9 May 2023 (UTC)[reply]

Can every number greater than 1 be written as the sum of a square and a prime? There are 5 squares less than 35. There are 11 primes less than 35. 5 × 11 = 55, which is more than 35. Yet the number 34 cannot be written as the sum of a square and a prime. --Lambiam 05:37, 10 May 2023 (UTC)[reply]

Sorry. Error on my part. I was just wrong when I calculated it. I thought the chances of a number being able to be expressed that way went down for sufficiently large numbers but they don't, they go up. NadVolum (talk) 09:34, 10 May 2023 (UTC)[reply]

Just did a Google search on power of two plus a prime as the same argument about the chances going up would apply there, got an interesting article

Cook, John (4 May 2020). "Prime plus power of 2 | Generalized Polignac conjecture". www.johndcook.com.

NadVolum (talk) 10:34, 10 May 2023 (UTC)[reply]

So I looked around online for some resource that might have an answer to this, and as it turns out someone asked a similar Math StackExchange question; in particular, they postulate that all primes greater than 2 (a subset of integers > 1 coprime to 6, ignoring 3) can be written as the sum of a 3-smooth number and a prime. No luck in that post either. GalacticShoe (talk) 23:40, 15 May 2023 (UTC)[reply]