Let’s have a circle with diameter D_1=5^2^{n-1} where n any non-zero natural number. The number of primitive right angle triangles which can be inscribed in this circle is equal to 2^{n-1}. Additionally, the number of primitive right angle triangles inscribed in a circle with diameter equal to a product of primes of the form p=a^2+b^2 is obtained from the diameter D_2=p_1∙p_2∙…p_n=2^{n-1} .

n Length of Diameter Length of Diameter Number of
5^2^{n-1} p_1∙p_2∙…p_n Primes

1 5^1
2 5^2 5∙13 2
3 5^4 5∙13∙17 3
4 5^8 5∙13∙17∙29 4
5 5^16 5∙13∙17∙29∙37 5

We can increase the numbers indefinitely; the ratio of the two diameters increases rapidly. Now we have to find primes whose product gives a ratio 5^2^{n-1}/{ p_1∙p_2∙…p_n}≈1. At this point I want to remind you that the primes are of the form p=a^2+b^2 and from both diameters we obtain equal numbers of primitive right angle triangles inscribed in the circles. In order to find primes in ascending order that give such ratios, we have to find the vicinity in which they lie as follows:
Let’s take the diameter 5^16. Five primes are required and 5^16=5^15∙5^1 and (5^{1/15}∙5)^3≈172. So we have found the vicinity in which these primes lie. Below are the diameters equal to the new products of primes and their vicinities..

Length of Diameter Length of Diameter Vicinity in which Number of Primes
D_1=5^2^{n-1} equal to the new product of primes these primes lie

5^4 5∙13∙17 same 3
5^8 13∙17∙29∙37 25 4
5^16 149∙157∙173∙181∙193 172 5
5^32 5317∙5323∙5327∙5339∙5353∙5363 5343 6
∙ ∙ ∙
∙ ∙ ∙
5^32768 5^2048 15

We can increase the exponents indefinitely. We know the vicinity we have to find the range in which the primes lie. Conjecturally I say, this range can be found as follows:
Vicinity /(1+1/Number of Primes) Vicinity ∙(1+1/ Number of Primes)
Let’s take the Vicinity 172:
172/(1+1/5)=143+1/3 172∙(1+1/5)=206+4/10
or 172^2=(143+1/3)( 206+4/10) and 206/2ln206-143/2ln143≈5
Is this conjecture correct?

Is there a way to include a Table in the text?

nb to Scott : the thread has not been closed...