[seqfan] Prime of the form 4*p + 3
zbi74583_boat at yahoo.co.jp
zbi74583_boat at yahoo.co.jp
Thu Jan 30 03:31:27 CET 2020
Hi Se fan [ Theorem ] If p and 4*p + 3 are both Prime then the following pairs satisfy the formula Sigma(m) = Sigma(n) = 3*m - n It is classified as C.2 .... See my past mail " Much easier generalized AP " I named it {3, -1} AP n, m : 2^3*5*{3*p, 4*p + 3} k*{u, v} is k*u, k*v 2^5*7*{3*p, 4*p + 3] 2^9*11*31*{3*p, 4*p + 3} 2^13*11*43*127*{3*p, 4*p + 3}
If p and 2*p + 1 are both Sophie German Prime then 4*p + 3 is Prime Does anyone know the proof that Prime of the form 4*p + 3 exist infinitely ?
prog(k) = { my(m = 90, u = 3, v = -1); until(k<m, my(n = (sigma(m) - u*m)/v) ; if(0<n, if(1/u*(sigma(n) - v*n) == m, print(factor(m), ",", factor(n)))); m++)};prog(8000000)[3, 1; 5, 1; 13, 1],[3, 1; 83, 1][2, 4; 31, 1],[2, 4; 31, 1][2, 3; 5, 1; 31, 1],[2, 3; 3, 1; 5, 1; 7, 1][2, 3; 11, 1; 19, 1],[2, 3; 3, 1; 59, 1][3, 2; 5, 1; 41, 1],[3, 2; 251, 1][2, 3; 5, 1; 47, 1],[2, 3; 3, 1; 5, 1; 11, 1][2, 1; 5, 1; 7, 1; 29, 1],[2, 1; 3, 1; 5, 1; 59, 1][3, 1; 7, 1; 11, 2],[7, 1; 13, 1; 37, 1][2, 3; 5, 1; 67, 1],[2, 7; 3, 1; 5, 1][2, 3; 5, 1; 71, 1],[2, 3; 3, 1; 5, 1; 17, 1][2, 3; 5, 1; 79, 1],[2, 3; 3, 1; 5, 1; 19, 1][2, 3; 5, 1; 103, 1],[2, 3; 3, 1; 5, 3][3, 1; 7, 1; 11, 1; 19, 1],[3, 1; 31, 1; 59, 1][3, 2; 7, 2; 11, 1],[3, 2; 17, 1; 37, 1][2, 3; 5, 1; 127, 1],[2, 3; 3, 1; 5, 1; 31, 1][2, 5; 7, 1; 23, 1],[2, 5; 3, 1; 5, 1; 7, 1][2, 1; 5, 1; 11, 1; 53, 1],[2, 1; 3, 1; 971, 1][2, 3; 5, 1; 151, 1],[2, 3; 3, 1; 5, 1; 37, 1][2, 3; 5, 1; 167, 1],[2, 3; 3, 1; 5, 1; 41, 1][2, 3; 5, 1; 191, 1],[2, 3; 3, 1; 5, 1; 47, 1]
If m = n then it is PN
Yasutoshi
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