[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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