A mathematical hypothesis

According to my hypothesis, any natural number, so any positive integer can be produced as the sum of the square of an integer and a prime or at most two prime.

In this case, it is important (otherwise the conjecture is not true) to include 1 as prime number. (My personal point of view is that 1 should be considered as prime, but keeping the necessary distinctions as appropriate: primes bigger than 1 or bigger than 2. Out of respect for the tradition, it is reasonable to call prime numbers the primes bigger than 1, big primes the primes bigger than 2, in the end absolute prime numbers the set of all primes and the 1. 1 merit the respect, not only because that it is the ancestor of all numbers, but because it fulfills perfectly the most natural definition of a prime number: p is prime number, if from a × b = p follows that a = 1 and b = p , or the opposite.)

Let us see some examples that obviously only confirm but do not prove the conjecture.

1 = 1 + 0²

2 = 1 + 1²

3 = 3 + 0²

4 = 3 + 1²

5 = 5 + 0²

6 = 5 + 1²

7 = 7 + 0²

8 = 7 + 1²

9 = 5 + 2²

10 = 1 + 3²

100 = 19 + 9²

1000 = 919 + 9²

10000 = 199 + 99²

123 = 2 + 11²

1234 = 1009 + 15²

12345 = 1109 + 106²

It is easy to spot that in some cases the number can be produced in different ways. For example:

12345 = 1109 + 106² = 2741 + 98²

Interestingly, most natural numbers can be produced as the sum of a square and one prime number, but there are “hard” cases that can only be solved with two prime numbers.

So far, according to calculations – partly reported by others – from 1 to 100 there are six exceptions.

25 = 5 + 19 + 1²

34 = 11+ 23 + 0²

58 = 3 + 53 + 1²

64 = 11 + 53 + 0²

85 = 31 + 53 + 1²

91 = 2 + 89  + 0²

The "difficult" cases between 100 and 1000: 121, 130, 169, 196, 214, 289, 324, 370, 400, 526, 529, 625, 676, 706, 771, 784, 841. It can be seen that their Frequency decreases fastly, but it is unlikely they will "run out". I suppose for example that the numbers of the form

2^2(2k+1)

are all difficult cases. (It would certainly be fitting to give a name to both types of natural numbers.)Was already made this hypothesis, I do not know. Among other things, that is why I did not want to give him a name (which is not important, but practical to mention it).

Moreover, I am convinced that this statement – unfortunately – cannot be proved in the same way as the famous Goldbach's conjecture (which, I think, has nothing to do with Gödel's "incompleteness" theorem, but is due to the mystery of the prime numbers).

This is our world.

March 3, 2019, Todor Simeonov

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