Problems & Puzzles: Puzzles

Puzzle 83.- The 6k+1 and 6k-1 boxes

As a result of a discussion about the distribution of the twin prime numbers sent to me by Alberto Zelaya (that will deserve later another puzzle than this) I made a numerical empirical search that conveyed to the following other issues:

Knowing that all the prime numbers can be dropped in one of the following boxes: the 6k+1 box or the 6k-1 box:

a) Are the prime numbers equally distributed in these two boxes?

Let P and Q to be the total (accumulated) number of prime numbers in the 6k+1 and 6k-1 boxes, respectively, for k=1 to certain K value. Empirically can be observed that it seems to be that P is never greater than Q. For k<315589197, I got that P=Q only in the following few cases: K=1, 2, 3, 6, 7, 13, 27, other wise P<Q.

I stopped the search when the status went at: K=315589197, P=46620532, Q=46624578, P+Q=93245110, (P-Q)/(P+Q)= -4.339101535726646e-5, abs(P-Q)=4046

b) Exist other K values such that P=Q?
c) Does this situation (P is never greater than Q) remain forever?
d) Is this behavior explainable, noteworthy or interesting?
e) Can the value (Q-P) be as large as desired?


Luis Rodríguez wrote (26/02/2000): 

"The puzzle related to Zelaya search is very interesting; Paulo Ribenboim explains its details in the p. 275 of "The new book of prime number records". According with this, the answers to the posted questions are:

a) Yes
b) Yes

c) No
d) Yes
e) Yes

The explanation about the large series of numbers needed to the inequalities to be inverted is related to the same random characteristic distribution of the primes which produces a kind of 'random walk', needing large quantity of steps for the required inversion . Please see also: "An Introduction to Probability Theory and Its Applications", Feller p. 83."

I (CR) have not the Feller's book, but at the p. 285 of the Ribenboim book we can read this:

"Tschebycheff noted already in 1853, that p3,1(x)< p3,2(x) and p4,1(x)< p4,1(x) for small values of x; in other words, there are more primes of the form 3k+2 than of the form 3k+1 (resp. more primes 4k+3 than primes 4k+1) up to x (for x not too large). Are these inequalities true for every x?…it may be shown that these inequalities are reversed infinitely often. Thus Leech computed in 1957 that x1=26861 is the smallest prime for which p4,1(x)> p4,1(x); see also Bays and Hudson (1978) who found that x1=608981813029 is the smallest prime for which p3,1(x)> p3,2(x)…On the average p3,2(x) - p3,1(x) is asymptotically (x^1/2)/(log x). However, Hudson (with the help of Schinzel) showed in 1985 that

limit  as xàinfinite (p3,2(x) - p3,1(x))/((x^1/2)/(log x))

does not exist (in particular, it is not equal to 1)"

So, after this precious paragraphs pointed out kindly by Rodríguez, we know now that for sure P will be equal to Q again infinitely times, remaining only the following question: when, for the first time, P will be Q again after k=27 in x=6*k+/-1?



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