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Old 2020-03-30, 11:52   #1
enzocreti
 
Mar 2018

1F016 Posts
Default 69660 and 92020

69660 and 92020 are multiple of 215 and congruent to 344 mod 559
92020=lcm(215,344,559)+69660


| denotes concatenation in base 10


2^69660-1 | 2^69559-1 is prime
2^92020-1 | 2^92019-1 is prime!!!

Last fiddled with by enzocreti on 2020-03-30 at 11:53
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Old 2020-03-30, 12:19   #2
LaurV
Romulan Interpreter
 
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Jun 2011
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please show us a proof that they are prime
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Old 2020-03-30, 12:50   #3
enzocreti
 
Mar 2018

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Default ...

Well...
actually they are only probable primes... maybe in future they will be proven primes

Last fiddled with by enzocreti on 2020-03-30 at 12:52
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Old 2020-03-30, 17:39   #4
enzocreti
 
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Quote:
Originally Posted by LaurV View Post
please show us a proof that they are prime


http://factordb.com/index.php?id=1100000001110801143

http://factordb.com/index.php?query=...%2B2%5E92019-1

Last fiddled with by enzocreti on 2020-03-30 at 17:55
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Old 2020-03-30, 23:06   #5
enzocreti
 
Mar 2018

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Default ... I note also...

I note also that


(lcm(215,344,559))^2=4999*10^5+69660-60


I note that the polynomial

X^2-X*429^2+7967780460=0 has the solution x=69660


If you see the discriminant of such polynomial you can see interesting things about pg primes with exponent multiple of 43

I note that 429^2 is congruent to 1 mod 215 and to 1 mod 344.



I note that 92020*2+1=429^2

The discriminant of the polynomial is 429^4-4*7967780460 which is a perfect square and lcm(215,344,559) divides 429^4-4*7967780460-1

Last fiddled with by enzocreti on 2020-04-01 at 08:07
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