Conjecture no. 39

If 2+3+5…+n is equal to a prime , then it is also a prime of the form x^2+1 where n is greater than two finitely often?

Some examples will follow

2+3= 2^2+1

2+3+5+7= 4^2+1

2+3+5+7+… 37 =14^2+1

It would be appreciated if someone could find more examples of this phenomenon.

Regarding the composites of the form 2+3+5+7+…n is there only finitely many which sum to triangular numbers. An example of this is 2+3+5=10 , 10 is triangular.

A triangular number is a figurate number that is analogous to the factorial i.e the factorial operation over the positive integers is defined as the multiplication of consecutive positive integers. Similarly the triangular number operation is defined as the successive adding of consecutive positive integers. One can also define a recursive formula for generating triangular numbers e.g n(n+1)|2.

As always I welcome the readers to prove or disprove this conjecture.

Thank you,

Nunghead