Numbers of the form 41s+r
 

Let be k an integer
I search the values of k such that:
1- k is a multiple of 43
2-k is of the form 41s+r where r can be one of these numbers: 1, 10, 16, 18, 37.
s in the integers obviously


Somebody can give me a routine for SAGE for exampel to find them out?

41*21 = 1 (mod 43) so 41^(-1) = 21 (mod 43):



41s+1:
---------
41s+1 = 0 (mod 43) => 41s = -1 (mod 43) => s=(-1)*41^(-1) = (-1)*21 = -21 = 22 (mod 43).
so s=22+43n and the numbers are then: 41*(43n+22)+1 = [U]1763n+903[/U]

41s+10:
---------
41s+10 = 0 (mod 43) => 41s = 33 (mod 43) => s=33*21 = 5 (mod 43)
so s=5+43n and the numbers are then: 41*(43n+5)+10 = [U]1763n+215[/U]

41s+16:
---------
41s+16 = 0 (mod 43) => 41s = 27 (mod 43) => s=27*21 = 8 (mod 43)
so s=8+43n and the numbers are then: 41*(43n+8)+16 = [U]1763n+344[/U]

41s+18:
---------
41s+18 = 0 (mod 43) => 41s = 25 (mod 43) => s=25*21 = 9 (mod 43)
so s=9+43n and the numbers are then: 41*(43n+9)+18 = [U]1763n+387[/U]

41s+37:
---------
41s+37 = 0 (mod 43) => 41s = 6 (mod 43) => s=6*21 = 40 (mod 43)
so s=40+43n and the numbers are then: 41*(43n+40)+37 = [U]1763n+1677[/U]


So 1763n + m, where m is 215,344,387,903,1677

congruent to 6 or 7 mod 13
 

And what if I search for values of k such that
k is multiple of 43
k is of the form 41s+r with r=1,10,16,18,37
then if k is even, k have to be congruent to 6 mod 13
if k is odd, k have to be congruent to 7 mod 13
?

[QUOTE=enzocreti;508421]And what if I search for values of k such that
k is multiple of 43
k is of the form 41s+r with r=1,10,16,18,37
then if k is even, k have to be congruent to 6 mod 13
if k is odd, k have to be congruent to 7 mod 13
?[/QUOTE]
:picard:
The Pari-GP function chinese() [named in honor of the Chinese Remainder Theorem, or CRT] is made for such things. The conditions 43|k, k == 1 (mod 41), 2|k, and k == 6 (mod 13) give

? chinese([Mod(0,43),Mod(1,41),Mod(0,2),Mod(6,13)])
%1 = Mod(23822, 45838)

while 43|k, k == 1 (mod 41), k == 1 (mod 2), and k == 7 (mod 13) give

? chinese([Mod(0,43),Mod(1,41),Mod(1,2),Mod(7,13)])
%2 = Mod(32637, 45838)

and similarly for the others.

[QUOTE=enzocreti;508421]And what if I search for values of k such that
k is multiple of 43
k is of the form 41s+r with r=1,10,16,18,37
then if k is even, k have to be congruent to 6 mod 13
if k is odd, k have to be congruent to 7 mod 13
?[/QUOTE]

41s+1: 22919n+903, 22919n+9718

41s+10: 22919n+215, 22919n+14319

41s+16: 22919n+344, 22919n+9159

41s+18: 22919n+10965, 22919n+19780

41s+37: 22919n+8729, 22919n+17544