A231201 Number of ways to write n = x + y (x, y > 0) with 2^x + y prime.

0, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 2, 1, 2, 4, 4, 4, 5, 3, 2, 4, 1, 2, 1, 4, 4, 4, 2, 3, 4, 4, 4, 3, 2, 5, 4, 4, 4, 3, 5, 4, 5, 3, 4, 7, 6, 5, 2, 5, 3, 5, 7, 1, 3, 5, 5, 4, 6, 5, 4, 4, 5, 3, 1, 4, 7, 6, 5, 5, 4, 5, 7, 4, 5, 3, 5, 6, 8, 3, 4, 4, 6, 3, 5, 2, 2, 3, 6, 6, 4, 5, 6, 5, 5, 8, 6, 4, 7, 5, 4

Offset: 1

Author

Zhi-Wei Sun, Nov 10 2013

Keywords

• nonn
• look

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, any integer n > 7 can be written as x + y with 0 < x < y such that 2^x + y is prime.

(ii) Every n = 2, 3, ... can be written as x + y (x, y > 0) with 2^x + y*(y+1)/2 prime.

(iii) Each integer n > 1 can be written as x + y (x, y > 0) with 2^x + y^2 - 1 prime. Also, any integer n > 1 not equal to 16 can be written as x + y (x, y > 0) with 2^x + y^4 - 1 prime.

We have verified part (i) of the conjecture for n up to 1.6*10^6. For example, 421801 = 149536 + 272265 with 2^149536 + 272265 prime.

We have extended our verification of part (i) of the conjecture for n up to 2*10^6. For example, 1657977 = 205494 + 1452483 with 2^205494 + 1452483 prime. - Zhi-Wei Sun, Aug 30 2015

The verification of part (i) of the conjecture has been made for n up to 7.29*10^6. For example, we find that 5120132 = 250851 + 4869281 with 2^250851 + 4869281 a prime of 75514 decimal digits. - Zhi-Wei Sun, Nov 16 2015

We have finished the verification of part (i) of the conjecture for n up to 10^7. For example, we find that 9302003 = 311468 + 8990535 with 2^311468 + 8990535 a prime of 93762 decimal digits. - Zhi-Wei Sun, Jul 28 2016

In a paper published in 2017, the author announced a USD $1000 prize for the first solution to his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Dec 03 2017

Examples

			a(8) = 1 since 8 = 3 + 5 with 2^3 + 5 = 13 prime.
a(53) = 1 since 53 = 20 + 33 with 2^{20} + 33 = 1048609 prime.
a(64) = 1 since 64 = 13 + 51 with 2^{13} + 51 = 8243 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Write n = k + m with 2^k + m prime, a message to Number Theory List, Nov. 16, 2013.
• Z.-W. Sun, On a^n+ bn modulo m, arXiv:1312.1166 [math.NT], 2013-2014.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2015.
• Z.-W. Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT].)

Crossrefs

Cf. A000040, A000079, A228425, A228428, A228429, A228430, A228431, A231516, A231555, A231557, A231561, A231577, A231631.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[2^x+n-x],1,0],{x,1,n-1}]
Table[a[n],{n,1,100}]