mersenneforum.org a quadratic residue modulo and( Mersenne numbre)
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 2020-09-28, 14:12 #1 baih     Jun 2019 2×17 Posts a quadratic residue modulo and( Mersenne numbre) (2n-2)-7 is a quadratic residue modulo M(n) x2 = ( 2n-2)-7 mod M(n) let n >5 example 52 --------------------------------= (27-2)-7 mod 27-1 38242----------------------------= (213-2)-7 mod 213-1 1812------------------------------= (217-2)-7 mod 217-1 4706212-------------------------= (219-2)-7 mod 219-1 13192077362------------------= (231-2)-7 mod 231-1 7558603523101899312------------------------------------------= (261-2)-7 mod 261-1 5375839232663506644937839732---------------------------= (289-2)-7 mod 289-1 560073960062182455573993569805432-------------------= (2107-2)-7 mod 2107-1 1180149151945109291147994277315908219182--------= (2127-2)-7 mod 2127-1
 2020-09-28, 14:23 #2 paulunderwood     Sep 2002 Database er0rr 3,533 Posts It is true for all odd prime p that kronecker(2^(p-2)-7,2^p-1) == 1. What is your point?
 2020-09-28, 14:31 #3 baih     Jun 2019 2×17 Posts why is true Is there an order of Distribution of quadratic residues
 2020-09-28, 14:40 #4 paulunderwood     Sep 2002 Database er0rr 3,533 Posts 2^(p-2)-7 has the same residue as 2^p - 7*4; same as 1 -7*4; same as -27; same as -3, and this is always true for odd Mersenne. LL is the same as x^(n+1)==1 mod (M_p, x^2-4*x+1) and the polynomial has discriminant 12 the kronecker symbol of which is -1; same as for 3. The kronecker symbol of -3 over M_p is 1 since kronecker(-1,M_p)==-1 because M_p==3 mod 4. That is how I see it. Last fiddled with by paulunderwood on 2020-09-28 at 14:51
 2020-09-28, 15:26 #6 paulunderwood     Sep 2002 Database er0rr 3,533 Posts Re-writing your terrible use of brackets... 2^(p-2)- 7 - ((a*(a+1)/2) - 1)*6 has the same symbol over M_p as: 2^(p-2) - 1 - 3*a*(a+1) same as 2^p -4 - 12*a*(a+1) same as -3 - 12*a*(a+1) same as -3(4*a^2+4*a+1) same as -3(2*a+1)^2 same as -3 I.e. the symbol is 1. Last fiddled with by paulunderwood on 2020-09-28 at 15:31
2020-09-28, 15:48   #7
baih

Jun 2019

1000102 Posts

Quote:
 Originally Posted by paulunderwood Re-writing your terrible use of brackets... 2^(p-2)- 7 - ((a*(a+1)/2) - 1)*6 has the same symbol over M_p as: 2^(p-2) - 1 - 3*a*(a+1) same as 2^p -4 - 12*a*(a+1) same as -3 - 12*a*(a+1) same as -3(4*a^2+4*a+1) same as -3(2*a+1)^2 same as -3 I.e. the symbol is 1.
thanks I am not a pure mathematician I am (just a fake mathematician)
my work is a programmer (java android)

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