Developments |
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| This decomposition is always possible except for p(1) = 2, p(2) = 3 and p(4) = 7 or p(n+1) >= (3/2) * p(n) holds only for n = 1, n = 2 and n = 4 (for the proof see arXiv:0711.0865). |
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| p is a prime such that p > 3 and p + 2 is also prime if and only if p has a weight equal to 3. (see arXiv:0711.0865). |
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| For p(n) different from 2, 3 and 7, we have : |
| l(n) = p(n) - g(n) = 2 * p(n) - p(n+1) = k(n) * L(n), |
| p(n) = l(n) + g(n) = k(n) * L(n) + g(n), |
| gcd(g(n),2) = 2, |
| gcd(p(n),g(n)) = gcd(p(n) - g(n),g(n)) = gcd(l(n),g(n)) = gcd(L(n),g(n)) = gcd(k(n),g(n)) = 1, |
| 3 <= k(n) <= l(n), |
| 1 <= L(n) <= l(n) / 3, |
| 2 <= g(n) <= k(n) - 1, |
| 2 * g(n) + 1 <= p(n). |
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