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The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime, and http://oeis.org/A005377 for a list of prmes not of the form $p+x^2$ with $p$ prime and $x$ a positive integer.

Concerning your second question on uniform explanations, you may consult Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017 for a General Hypothesis on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.

The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime, and http://oeis.org/A065377 for a list of primes not of the form $p+x^2$ with $p$ prime and $x$ a positive integer.

Concerning your second question on uniform explanations, you may consult Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017 for a General Hypothesis on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.

The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime.

Concerning your second question on uniform explanations, you may consult Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017 for a General Hypothesis on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.

The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime, and http://oeis.org/A005377 for a list of prmes not of the form $p+x^2$ with $p$ prime and $x$ a positive integer.

Concerning your second question on uniform explanations, you may consult Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017 for a General Hypothesis on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.

The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime.

I don't think it is easy to pose new nice conjecures on primes. For a general hypothesis on representations involving primes, see Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017.

The conjecture PSQ is essentially not new. In 1923 Hardy and Littlewood [Acta Math. 44(1923), 1-70] conjectured that every large integer, not being a square, may be expressed as the sum of a prime and a square. See also http://oeis.org/A020495 for the list of non-square positive integers which are not of the form $p+x^2$ with $p$ prime.

Concerning your second question on uniform explanations, you may consult Conjecture 2.1 of my paper Conjectures on representations involving primes published in 2017 for a General Hypothesis on representations involving primes.

PS: I don't think it is easy to pose new nice conjecures on primes.