Can you prove or disprove the following list of my conjectures?

Question

The following three statements are my own conjectures, not a homework problem.

$a)$ For $n = 3, 4, 5,..$, every square integer $n^2$ can be expressed as the sum of a prime $p$ and two other primes $q$ and $r$ multiplied together with $q, r < n$.

$b)$: (Similar to $a)$) Every positive integer $n>10$ can be written as a sum of a prime $p$ and two other primes (not necessarily distinct) $q$ and $r$ multiplied together.

$c)\,\mathbf{[proven]}$: The digital root of every perfect number except $6$ is $1$.

Can you prove or disprove them? If this is difficult, are there any implications between $a),b)$ and Goldbach's conjecture?

Answer 1

The conjecture (c) as written is false. $n=2^{p−1}(2^p−1)$ is a perfect number for $p=2203,$ but the sum of the digits of the sum of the digits of $n$ is 19.

However, if you keep iterating you arrive at 10 it seems. In other words, the digital root seem to be 1.

Actually, this is correct (all even perfect numbers other than 6 have digital root 1) and has been proven in http://apfstatic.s3.ap-south-1.amazonaws.com/s3fs-public/09-comac_digital_roots_of_perfect_numbers%20(1).pdf.

Answer 2

(b) is an open problem.

If it weren't, then we would have resolution of both Lemoine's conjecture$^{[1]}$ and Marnell's prime and semiprime conjecture$^{[2]}$ (for which no exceptions below $1 \times 10^8$ exist).

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[1] https://en.wikipedia.org/wiki/Lemoine%27s_conjecture

[2] Geoffrey R. Marnell, "Ten Prime Conjectures", Journal of Recreational Mathematics 33:3 (2004-2005), pp. 193--196.