cp's OEIS Frontend

A210844 A pair of solutions of a congruence related to A141453.

%I A210844 #10 Sep 17 2012 12:24:53
%S A210844 3,5,9,15,33,63,255,513,16383,131073,262143,1048575,4294967295,
%T A210844 4611686018427387903,1237940039285380274899124223,
%U A210844 324518553658426726783156020576255,340282366920938463463374607431768211455
%N A210844 A pair of solutions of a congruence related to A141453.
%C A210844 See the comment on A141453. There r(a(n)) is the present a(n).
%C A210844 The next entry a(18) has 158 digits.
%C A210844 The sequence of exponents of 2 of the Fermat and Mersenne primes FM:=A141453 (including the prime 2) starts with k:=[0, 1, 2, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127, 521,...], n>=1.
%C A210844 For the second k entry one can also take 2 instead of 1. Then a(2) should be replaced by 7.
%C A210844 a(n) and FM(n)*2^(k(n)+1) - a(n) are an incongruent pair of solutions of the congruence x^2 == 1 (mod FM(n)*2^(k(n)+1)), n>=1. For n>=3 there are all-together eight incongruent solutions. The trivial pair of positive solutions is always 1 and FM(n)*2^(k(n)+1) - 1. Two more pairs should therefore be found.
%F A210844 a(n) = sqrt(FM(n)*2^(k(n)+2) + 1), n>=1, with FM(n):=A141453(n) and the sequence k is given for n=1..18 in the comment section.
%e A210844 From Wolfdieter Lang, Apr 10 2012 (Start)
%e A210844 a(1)=3 because 3^2 = 9 == 1 (mod 2*2^(0+1)) = 1 (mod 4). The incongruent companion solution is 4 - 3 = 1. This is the trivial pair of solutions.
%e A210844 a(2)=5 because 5^2 = 25 == 1 (mod 3*2^(1+1)) = 1 (mod 12). The incongruent companion solution is 12 - 5 = 7, obtained also by taking k(2)=2. The trivial pair of solutions is (1,11).
%e A210844   1, 5, 7 and 11 are all the solutions of this congruence.
%e A210844 a(3)=9  because 9^2 = 81 == 1 (mod 5*2^(2+1)) = 1 (mod 40).
%e A210844   The companion solution is 40 - 9 = 31. The trivial pair is (1,39). The missing two pairs are (11,29) and (19,21), and all eight incongruent solutions are 1, 9, 11, 19, 21, 29, 31 and 39.
%e A210844 (End)
%Y A210844 Cf. A141453.
%K A210844 nonn
%O A210844 1,1
%A A210844 _Wolfdieter Lang_, Mar 28 2012