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For any integer m, define s(m) as the smallest positive integer s such that for each n=s,s+1,... there are primes p>q>2 with p+(1+(n mod 2))q=n and ((p-(1+(n mod 2))m)/q)=((q+m)/p)=1. If such a positive integer s does not exist, then we set s(m)=0.

Zhi-Wei Sun has the following general conjecture: s(m) is always positive. In particular, s(0)=1239,

s(1)=1470, s(-1)=2192, s(2)=1034, s(-2)=1292,

s(3)=1698, s(-3)=1788, s(4)=848, s(-4)=1458,

s(5)=1490, s(-5)=2558, s(6)=1115, s(-6)=1572,

s(7)=1550, s(-7)=932, s(8)=825, s(-8)=2132,

s(9)=1154, s(-9)=1968, s(10)=1880, s(-10)=1305,

s(11)=1052, s(-11)=1230, s(12)=2340, s(-12)=1428,

s(13)=2492, s(-13)=2673, s(14)=1412, s(-14)=1638,

s(15)=1185, s(-15)=1230, s(16)=978, s(-16)=1605,

s(17)=1154, s(-17)=1692, s(18)=1757, s(-18)=2292,

s(19)=1230, s(-19)=2187, s(20)=2048, s(-20)=1372,

s(21)=1934, s(-21)=1890, s(22)=1440, s(-22)=1034,

s(23)=1964, s(-23)=1322, s(24)=1428, s(-24)=2042,

s(25)=1734, s(-25)=1214, s(26)=1260, s(-26)=1230,

s(27)=1680, s(-27)=1154, s(28)=1652, s(-28)=1808,

s(29)=1112, s(-29)=1670, s(30)=1820, s(-30)=1284.

Note that s(1)=1470 means that a(n)>0 for all n=1470,1471,... That s(0)=1239 is related to a conjecture of Olivier Gérard and Zhi-Wei Sun.

If we replace ((p-1-(n mod 2))/q)=((q+1)/p)=1 in the definition of a(n) by ((p-1)/q)=((q+1)/p)=1, then the new a(n) seems positive for any n>1181.

Conjecture: a(n)>0 for all even n>=8070 and odd n>=18680.

This conjecture unifies the twin prime conjecture, Goldbach's conjecture and Lemoine's conjecture. It has been verified for n up to 10^7.

Zhi-Wei Sun also made the following conjecture: Any integer n>=6782 can be written as p+q+(n mod 2)q with p>q and p, q, q-6, q+6 all prime, and any integer n>=4410 can be written as p+q+(n mod 2)q with p>q and p, q, 2q-3, 2q+3 all prime, and any integer n>=16140 can be written as p+q+(n mod 2)q with p>q and p, q, 3q-2, 3q+2 all prime.

Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.

This has been verified for n up to 2*10^6.

Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.

Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.

Conjecture: for all n > 3, a(n) > 0.

If 2n = p + q and p+6 is also a prime, 2n+6 can be written as the sum of two primes p+6 and q.

The conjecture is weaker than a conjecture of Sun posed in 2012 (see A219055). - Zhi-Wei Sun, Mar 18 2015