Problems & Puzzles:
Puzzles
Puzzle 223.
Consecutive primes and powers of 2
How often the sum of two consecutive
primes p & q is equal to a power o 2?
p + q = 2^{n}
As you know without even thinking, the earliest solution is 3 + 5 = 2^{3}
. We will be represent any solution just by the pair (n, p).
Here are the first five solutions as reported
very recently by Naohiro Nomoto
at A071087 :
n 
p 
3 
3 
7 
61 
13 
4093 
77 
? 
182 
? 
I have calculated two more entries this
week:
*
I have not shown on purpose some of the
primes p in both Tables because probably you'll be interested in
recalculating them by your own.
Question 1:
Find the first n such that p is titanic.
Question 2: In the
meanwhile, find
the p value for
n=1821
Solution:
Contributions came from Jason
Earls, David Terr, Ken Wilke and J. K. Andersen.
Jason, David and Ken
solved the question 2. Andersen solved both questions.
As a matter of fact Jason Earls
found all the primes not shown in the Tables. David Terr found p
for n=1821 "with
Mathematica in 5 seconds".
Ken used a theoretical
shortcut for getting p for n=1821, that may be of interest for getting
larger solutions:
Let p and q be consecutive primes such that p < q
and p + q = 2^(n+1) for some integer n > 2. Then since p < (p + q)/2 = 2^n
< q, we must have p = 2^n  k and q = 2^n + k for some integer k.
Considering p, q and 2^n (mod 6) we have: for n odd,
2^n = = 4, p = = 1 and k= =3, all (mod 6) and for n even, 2^n = = 2, p = =
5 and k= =3, all (mod 6). Then it is easy to construct a program in UBASIC
to check possible values of k for a given value of n.
For n = 1821, I chose to examine gcd (V, T) where V
is the product of all primes < 1000 and T = pq = 2^3640  k^2 to get a
list of possible values of k to check for primality. Only values of k in
which gcd(V, T) = 1 need to be checked. In addition, I used that fact that
3 is a quadratic residue of all primes of the form 6d + 1 and a quadratic
nonresidue of all primes of the form 6d + 5. Checking possible values of
k < 200 , I found that only k = 663 needed to be checked. UBASIC's
APRTCLE program verified that both p = 2^1820  663 and q = 2^1820 + 663
are primes. Similarly, for n = 1100 p = 2^1099  1035 and q = 2^1099 +
1035; for n = 77
p = 2^76  15 and q = 2^76 + 15 and for n = 182 p =
2^181  165 and q = 2^181 + 165.
My code follows:
4 J = 1: S = 1: T = 1
5 while J < 1000
6 T = T * S
8 S = nxtprm(J)
10 J = S
12 wend
14 print T
20 M = 2^1820: N = 0
21 while N < 500
22 K = 6 * N + 3:A = M + K: B = M  K
23 V = A * B
24 if gcd ( V, T) > 1 then 39
26 if kro (3, A) < > 1 then 39
28 if kro (3, B) < > 1 then 39
30 if modpow ( 3, A  1, A) = 1 then 32 else 39
32 if modpow ( 3, B  1, B) = 1 then 35 else 39
35 print N, K
39 inc N
40 wend
45 end.
If the value of M in line 20 is odd then lines 26
and 28 need to be changed to
26 if kro (3, A) < > 1 then 39
28 if kro (3, B) < > 1 then39
Andersen found the asked
solution to Q.1 this way:
I first sieved to 2^31 with my own C program.
Based on prp tests performed with PrimeForm/GW, the
first probable solution after n=1821 is n=9230 with titanic
p=2^92292211.
Note: Smaller n have only been eliminated by finding
the smallest k giving a probable prime p=2^n/2k (largest prp with n1
bits). In all cases, 2^np = 2^n/2+k was composite. These p have not been
proved prime and if one p is really composite then there could
theoretically be an unknown solution with a larger k giving a true prime.
The solution prp's 2^92292211 and 2^9229+2211 have 2779 digits. They have
passed many prp tests and could be proved prime with Marcel Martin's Primo
in around a week but I will not do it. Based on heuristics, I expect an
infinite number of solutions. There are no other solutions with n<10261.
As a curio, I encountered a big prime gap: From 2^1009380445 to
2^10093+4185. This is 12 times the expected gap size 6996.
Question 2
For n=1821: p=2^1820663. The primes 2^1820663 and
2^1820+663 with 548 digits have been proved with Primo.
...
From the prime number theorem, the probability a number around n is
prime is approximately 1/ln(n) where ln is the
natural logarithm. The average gap size around
n=2^10093 is then ln(n) = ln(2^10093) =
10093*ln(2) ~= 6996. This is what I call "the
expected gap size"...
It was tested that there are no primes between 2^92292211 and
2^9229+2211. I found "near" solutions where
2^n/2k was the largest prime below 2^n/2, and
2^n/2+k was also prime, but not the smallest prime above 2^n/2.
I will report my solution to EIS with a comment that it is only based
on prp's.
***
