A318202 Prime numbers p such that p-3 cannot be written as sum of two twin prime numbers (not necessarily forming a pair of twin primes).

2, 3, 5, 7, 97, 101, 521, 787, 907, 911, 1117, 1151, 1361, 3251, 4211

Offset: 1

Author

Dimitris Valianatos, Aug 21 2018

Keywords

• nonn
• more

Comments

Conjecture 1: All prime numbers >= 11 can be written as sum of 3 twin prime numbers.

Conjecture 2: All prime numbers aside from the 15 terms given here can be written as a sum of three twin prime numbers with (at least) one of them equal to 3.

If the sequence is finite then there are infinitely many twin primes.

For the terms in this sequence, the lexicographically least partitions into three twin primes are: 97=5+19+73, 101=11+17+73, 521=11+197+313, 787=5+139+643, 907=5+19+883, 911=11+17+883, 1117=5+19+1093, 1151=11+107+1033, 1361=11+29+1321, 3251=11+71+3169, 4211=11+41+4159.

a(16) > 10^6 if it exists. - Amiram Eldar, Dec 06 2018

Examples

			a(6) = 101 because 101 - 3 = 98 and (98 - 73 = 25, 98 - 71 = 27), (98 - 61 = 37, 98 - 59 = 39), ..., (98 - 5 = 93, 98 - 3 = 95) aren't twin primes.
		

Links

• David A. Corneth, PARI prog

Crossrefs

Cf. A000040, A001097, A007534, A129363, A321221.

Programs

Mathematica

p = Prime[Range[600]]; p2 = Select[p, PrimeQ[# - 2] || PrimeQ[# + 2] &]; Select[ p - 3, IntegerPartitions[#, {2}, p2] == {} &] + 3 (* Amiram Eldar, Nov 15 2018 *)

PARI

{forprime(n=2,10^4,p=n-3;forprime(t1=2,n,forprime(t2=t1,n,t12=t1+t2; if((isprime(t1-2)||isprime(t1+2))&&(isprime(t2-2)||isprime(t2+2)), if(t12==p,break(2)))));if(t12==2*n,print1(n", ")))}

PARI

isok(p) = {if (isprime(p), p -= 3; forprime(q = 2, p, if (isprime(r=p-q), if ((isprime(r+2) || isprime(r-2)) && (isprime(q-2) || isprime(q+2)), return (0)););); return (1));} \\ Michel Marcus, Dec 05 2018

PARI

\\ See Corneth link \\ David A. Corneth, Dec 05 2018

Extensions

2,3,5,7 prepended by David A. Corneth, Dec 05 2018