CRGreathouse said:
Well, then, start researching it to see who first asked it. Personally, I don't think it needs a name
First of all, Happy Thanksgiving everybody.
Second, a little bit of research to share with any who find the recent discussion to be of interest, and also to demonstrate the relevancy of the question: "
Are all 2n constructible as either the sum or difference of a twin prime and another integer with less than or equal to 2 divisors?"...
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via Wolfram Mathworld
Twin Primes
It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are 2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, ... (Sloane's A007534; Wells 1986, p. 132).
http://mathworld.wolfram.com/TwinPrimes.html
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via OEIS
A007534 Even numbers which are not the sum of a pair of twin primes.
2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518, 784, 786, 788, 904, 906, 908, 1114, 1116, 1118, 1144, 1146, 1148, 1264, 1266, 1268, 1354, 1356, 1358, 3244, 3246, 3248, 4204, 4206, 4208
No other n < 10^9. - T. D. Noe, Apr 10 2007
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 132.
EXTENSIONS
Conjectured to be complete (although if this were proved it would prove the "twin primes conjecture"!)
http://oeis.org/A007534
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Since CRGreathouse has already checked the question to 100,000,000, then the answer is "yes, at least to 10^9, and unknown thereafter.".
Thirdly, the question: "
Are all 2n constructible as either the sum or difference of two twin primes?" might reasonably be referred to as "The Extended Wells Conjecture" (unless Wells was merely passing on someone else's conjecture, which I have not been able to determine...) The question I initially asked, which, in the manner it is presented, allows one to segment the even numbers into partitions of "length" equal to the gap between the arithmetic average of consecutive twin primes...
e.g.
======================================
For...
z_n = p +/- 1, where p is a twin prime
q' = 1 or a prime number
Then...
# of 2n to 1000
not constructible as:
z_(n) + q'
or z_(n+1) - q'
for 2n > z_(n) and 2n < z_(n+1)
(Twin Prime Pair "Gap" = 1)
= 8
488, 496 [=(31^2 + 31)/2], 686 [=26^2], 694, 724, 746, 776, 784 [=28^2]
# of 2n to 1000
not constructible as either:
z_(n) + q'
or z_(n+1) - q'
for 2n > z_(n) and 2n < z_(n+1)
z_(n) + q'
or z_(n+2) - q'
for 2n > z_(n) and 2n < z_(n+2)
(Twin Prime Pair "Gap" = 2)
= 0
======================================
... could be construed as a corollary to that, one I have playfully dubbed the "Qis-Qin + p Corollary" to "The Extended Wells Conjecture." "Qis-Qin" is pronounced "Kiss Kin," suggestive of twin primes, and the letters stand for "
q*i^Squared + p" and "
q*i^Null + p." Thus, -q + p and +q + p.
Important to note is that a) The "The Wells Conjecture" could be wrong, but "The Extended Wells Conjecture" could be right, and b) Both could be wrong, in which case we would, by default, be back at the original question: Are all 2n constructible as either the sum or difference of a twin prime and another integer with less than or equal to 2 divisors? In such a case, then "The Qis-Qin + p Corollary" (Personal Nickname: "The Kissing Kin Corollary") would become the "The Qis-Qin + p Conjecture," unless, of course, the question has already been asked and given a name by another or a better name were to be suggested.
Question: Does anybody have a copy of "The Penguin Dictionary of Curious and Interesting Numbers"? (1986) If so, would they be kind enough to take a look at p.132 and see if it was Wells who came up with the conjecture discussed in this post or if he was just reporting the findings of someone else?Raphie
P.S. Fourthly, I want to make the point that the question I initially asked was based on a simple construction of 2n up to 488 (The lower bound on the answer to this question, yay or nay, with the assistance of CRGreathouse, T.D. Noe, digital computing and digital access to information has been upped by a factor of over 2 million in just a couple of days...). The point being that (I believe) one can ask meaningful mathematical questions, especially in the Information Age, based upon very small sample sizes, not incongruent, in spirit, with research conducted by Usability Expert Jacob Nielsen:
Jakob Nielsen's Alertbox, March 19, 2000
Why You Only Need to Test with 5 Users
http://www.useit.com/alertbox/20000319.html
P.P.S.
@ CRGreathouse: Thanks for the nudge to research this further. I'd like to note that what I am calling "The Wells Conjecture," in essence, "predicts" both the truth of "The Goldbach Conjecture" and the truth of "The Twin Prime Conjecture," and it does so with a subset of the primes, the relative scarcity of which is determinable via Brun's Constant. If one were able to exploit some logical loophole (which I suggest only in the abstract, not as a likelihood) in order to prove it in isolation (as an existence theorem), then one would also, by extension, be proving two of the most famous mathematical conjectures. Such proof would also, I might add, answer the question I asked in the affirmative. I personally view such manner of conjecture as may or may not have originated with Wells, even if not the original question that led me to it, as worthy of having a name.
