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Decomposition of an even number into two prime numbers

Posted 2 months ago
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Hypothesis 1: any even number greater than 2 can be decomposed into two prime numbers.

Hypothesis 2: for any number K> 3 there exists at least one R> 0 such that K-R = P and K + R = Q, where P, Q are prime numbers.

From the second hypothesis, the first follows automatically, since if we add both equalities, we get 2 * K = P + Q, that is, an even number on the left.

I drew a graph-polygon, which displays the dependence of K on the number of suitable R for a given number K.

And it turns out that the number of R increases with increasing K.

n = 10^4;
res = {};
ProgressIndicator[Dynamic[k], {4, n}]
For[k = 4, k < n, k++,
  c[k] = 0;
  For[r = 1, r < k, r++,

   If[PrimeQ[k + r] && PrimeQ[k - r],

     c[k] = c[k] + 1;
     ];
   ];
  res = Join[res, {{k, c[k]}}];
  ];
Graphics[Polygon[res], {Axes -> True}]

dependence of K on the count of R

Question. How to prove that the count of such R is constantly increasing?

7 Replies

There is fluctuation (so that value is not monotonically increasing). As for the rest, I'd recommend to start by placing Hypothesis 1 in a Google search.

yes, but the lower limit of these fluctuations only grows

That is not actually the case.

res[[32 ;; 64]]

(* Out[879]= {{35, 5}, {36, 6}, {37, 4}, {38, 5}, {39, 7}, {40, 4}, {41, 
  4}, {42, 8}, {43, 4}, {44, 4}, {45, 9}, {46, 4}, {47, 4}, {48, 
  7}, {49, 3}, {50, 6}, {51, 8}, {52, 5}, {53, 5}, {54, 8}, {55, 
  6}, {56, 7}, {57, 10}, {58, 6}, {59, 5}, {60, 12}, {61, 3}, {62, 
  5}, {63, 10}, {64, 3}, {65, 7}, {66, 9}, {67, 5}} *)

Notice that positions 37, 40, 41, 43, 44, 46, 47 are all "lows" (they are all surrounded by values greater-equal to 4). But positions 49, 61, and 64 are all lower still, at 3.

What you want to show, I guess, is that there is an asymptotic lim-inf and lim-sup curve, and both are in some sense "nice" (maybe O(n/logn^2) for example). Proving something along those lines would take real work (as in, it has not been done).

Proving something along those lines would take real work (as in, it has not been done).

and what needs to be done to prove that such an assimptot exists? The lower limit clearly exists, but I don `t know how to solve such a complex task.

Posted 1 month ago

Imagine the two primes as something physical, like the length of your shadow and your physical height. When you add those two numbers together there's a similar domain to work in. There are also some strange mathematical coincidences when we look at how those two "primes" would be "constructed".. but it's not likely this is the correct approach.

What this problem says to me is that the prime numbers are likely based on a purely additive theory of numbers with fewer rules and axioms. The solution should likewise use as few rules and axioms as possible to produce a satisfying answer.

Hypothesis 1: any even number greater than 2 can be decomposed into two prime numbers.

Really?

Any even Number has a factor 2 , which is prime. Look at 28. This is even, and

28 = 2 * 14

2 is prime, 14 not !???

28 = 2 * 14

but why the work is considered, there was a sum of two primes, for example, 28 = 17 + 11

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