Thread: 69660 and 92020
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Old 2019-12-18, 08:26   #2
enzocreti
 
Mar 2018

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Default 69660, 92020, 541456

69660, 92020 and 541456 are 6 mod 13 (and 10^m mod 41)




69660, 92020 and 541456 are multiple of 43


is there a reason why




(69660-6)/26 is congruent to 13 mod 43
(92020-6)/26 is congruent to 13 mod 43
(541456-6)/26 is congruent to 13 mod 43?


215, 69660, 92020, 541456 are multiple of 43

let be log the log base 10

int(x) let be the integer part of x so for example int(5.43)=5

A=10^2*log(215)-215/41

int(A)=227=B

215 (which is odd) is congruent to B+1 mod 13
69660 which is even is congruent to B mod 13
92020 is congruent to B mod 13
and 541456 is also congruent to B mod 13






215 (odd) is congruent to -1215 mod 13
69660 (even) is congruent to 1215 mod 13
92020 (even) is congruent to 1215 mod 13
541456 (even) is congruent to 1215 mod 13










215, 69660, 92020, 541456 can be written as 13x+1763y+769


(541456-769-13*93)/1763=306


(69660-769-13*824)/1763=33


(92020-769-13*781)/1763=46


as you can see 306,33 and 46 are all 7 mod 13

769+13*824
769+13*781
769+13*93 are multiples of 43


(69660-(10^3+215))/13=5265 which is 19 mod 43
(92020-(10^3+215))/13=6985 which is 19 mod 43
(541456-(10^3+215))/13=41557 which is 19 mod 43


10^3+215 is 6 mod 13


now
69660/13=5358,4615384...
92020/13=7078,4615384...
541456/13=41650,4615384...


the repeating term 4615384 is the same...so that numbers must have some form 13s+k?


215 (odd) is congruent to 307*2^2-10^3 or equivalently to - (19*2^6-1) mod 13
69660 (even) is congruent to 307*2^2-1001 or equivalenly to (19*2^6-1) mod 13
92020 (even) the same
541456 (even) the same




(19*2^6*(541456-92020)/(13*43)+1)/13+10=75215 is a multiple of 307=(215*10-1)/7=(5414560-1)/17637




pg(51456) is another probable prime with 51456 congruent to 10^n mod 41


75215=(51456*19+1)/13+10=(19*2^6*(541456-92020)/(13*43)+1)/13+10




215 is congruent to -1215=5*3^5 mod 13

69660 is congruent to 1215 mod 13 and so also 92020 and 541456


1215=(51456/2-2*13*10^2-43)/19




...so summing up...


215 (odd) is congruent to 3*19*2^2 mod((41*43-307)/(7*2^4)=13) where 41*43-307 is congruent to 10^3+3*19*2^2 mod (3*19*2^2=228)
69660 (even) is congruent to (3*19*2^2-1) mod 13 ...
the same for 92020 and 541456


another way is 215 (odd) is congruent to -15*81 mod ((41*43-307)/(307-15*13))
69660 (even) is congruent to 15*81 mod ((41*43-307)/(307-15*13))
the same for 92020 and 541456




215 is congruent to (41*43-307-10^3)/2 mod ((41*43-307)/(7*2^4))
69660 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4))
92020 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4))
541456 is congruent to (41*43-307-10^3)/2-1 mod ((41*43-307)/(7*2^4))




215 is congruent to 3*19*2^2 mod ((41*43-307)/(2*(19*3-1)))
69660 is congruent to 3*19*2^2-1 mod ((41*43-307)/(2*(19*3-1)))
and so 92020 and 541456




215 is congruent to 3*19*2^2 mod ((3^6-1)/(3*19-1))
69660 is congruent to 3*19*2^2-1 mod((3^6-1)/(3*19-1)) and so 92020 and 541456

51456 (pg(51456) is probable prime and 51456 is 10^n mod 41) is congruent to 19*3*2^4 mod 13

Pg(2131) is probable prime
2131 is prime
227=2131-307*4-26^2
So 215 is congruent also to 2131-307*4-26^2 mod 13
And 69660 is congruent to 2131-307*4-26^2+1 mod 13 and so also 92020 and 541456


69660 is congruent to 1763-307*5-1 mod ((1763-307)/112)=13)

And so 92020 and 541456

215 which is odd is congruent to - 1763+307*5+1 mod 13
215+1763-307*5-1 is divisible by 17 and by 13
And also 541456-1763+307*5+1 is divisible by 13 and 17
215 and 541456 have the same residue 10 mod 41

((541456-1763+307*5+1)/(13*17)+1) *200+51456=541456


69660 92020 541456 are congruent to 7*2^6 mod 26
7*2^6-(1763-307*5-1)=221=13*17
215+7*2^6 is a multiple of 221
541456-7*2^6 is a multiple of 221

Last fiddled with by enzocreti on 2020-01-26 at 17:58
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