I have been focusing on another interesting problem (https://www.ibenedek.com/post/extension-of-the-problem-of-a-b-in-n-2-n-1-2-1-then-a-b-is-unique) for a while, and I recognized if we build up a number pyramid of an equilateral triangle starting from 1 to n, every line of the triangle will contain the element from n^2+1 to (n+1)^2.
So, I was thinking on the proof of every line should contain at least one Prime.
The ChatGPT said the Bertrand's Postulate implies it. ChatGPT provided a wrong deduction; let's see why:
As you see, the ChatGPT made a mistake; it compared the (n+1)^2 with the 2n instead of the 2*n^2.
However, when I asked it to visualize the functions of [(n+1)^2, 2*n^2], it provided a correct Python code:
As you can see, the 2x^2 is getting bigger than the (x+1)^2 function very early.
If we check the following screenshot, we will see that the 2(x^2+1) is growing extremely fast compared to the functions of x^2+1 and (x+1)^2.
Let's A = (x+1)^2-(x^2+1) and B= 2(x^2+1)-(x+1)^2, hence its clearly visible that, let's introduce this operator : (<< := much much slower)
A << B
So, basically, the probability of finding a prime in [(x+1)^2,..., (x^2+1)] is much much smaller than finding it in [2(x^2+1),...., (x+1)^2].
And seeing the difference between A and B implies that it will be hard to find proof of having at least one prime in [(x+1)^2,..., (x^2+1)].
We need to recognize that, and we would like to prove that we always have a prime in every row of the following number pyramid:
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