... I note also...
69660 I note also that
(lcm(215,344,559))^2=4999*10^5+6966060
I notice that lcm(215,344,559)=22360
22360/(18*18)=69.01234567...
curious
I notice also that 541456 (multiple of 43),is  215 mod (18*181)
and 69660 (multiple of 43) is 215 mod (18*181)
I note that the polynomial
X^2X*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 429^44*7967780460 which is a perfect square and lcm(215,344,559) divides 429^44*79677804601
Pg(331259) is prime and pg(92020) is prime.
92020+(92020/2151)*559+546=331259
Again magic numbers 559 and 546 strike!
So 331259 is a number of the form
215*(13s1)+(13*b2)*559+546
For some s, b positive integers
So there are pg primes pg(75894) and pg(56238) with 75894 and 56238 multiple of 546 and pg(331259) with 331259 of the form 215m+559g+546 for some positive m and g.
Pg(69660) is prime. 69660=(3067*8546)*311# where # is the primorial and 3067 is a prime of the form 787+456s
notice that lcm(215,344,559)=22360
22360/(18*18)=69.01234567...
curious
I notice also that 541456 (multiple of 43),is  215 mod (18*181)
and 69660 (multiple of 43) is 215 mod (18*181)
So we have pg(215) is prime pg(69660) is prime pg(92020) is prime
With 215 69660 and 92020 multiple of 43
69660=215+(18*181)*215
92020=69660+18*18*69+4
215 is congruent to 108 mod (18*181=323)
541456 is congruent to 108 mod (18*181)
92020 is congruent to (17*171) mod (18*181)
69660 is congruent to 215 mod (18*181)
108=6^318^2
17*171+36216=17*171+6^26^3=108
215=6^31
To make it easier
215, 69660, 541456 are congruent to plus or minus 215 mod 323
92020 is congruent to (17*171) mod (18*181)
curious that 289/215 is about 1.(344)...
and 541456 92020 69660 215 are congruent to plus or minus 344 mod 559
215, 69660, 541456 are congruent to plus or minus 6^31 mod 323
92020 is congruent to (12/9)*6^3 mod 323
92020*9/12 is congruent to 6^3 mod 323
92020 is congruent to (17^21) mod 323 and to  (6^21) mod 323
92020 is a number of the form 8686+13889s
13889=(6^3+1)*64
215 69660 92020 541456 are + or  344 mod 559
lcm(215,344,559)86*(10^2+1)+6^31=(6^3+1)*2^6+1
92020=69660+lcm(215,344,559) so you can substitute
92020=69660+86*(10^2+1)6^3+1+(6^3+1)*2^6+1
86*(10^2+1) mod 323 is 17^21
215, 69660, 541456 are multiple of 43 and congruent to 10 and 1 mod 41
They are congruent to plus or minus 215 mod 323
92020 is congruent to 2^4 (not a power of 10) mod 41
92020 is congruent to (2^4+1)^21 mod 323
288 is 17^21
288 in base 16 is 120
120=11^21
also 323=18^21 in base 16 is 143=12^21
344*((14444561763*2^9) /3441)=541456
1444456=lcm(13,323,344)
541456=lcm(13,323,344)344*(41*2^6+1)
215 69660 92020 541456 are congruent to plus or minus (3^a*2^b) mod 323
215 is congruent to  108 mod 323
541456 is congruent to 108 mod 323
69660 is congruent to  108 mod 323
92020 is congruent to 288 mod 323
108 and 288 are numbers of the form 3^a*2^b
So exponents multiple of 43 are congruent to plus or minus 344 mod 559 and to plus or minus a 3smooth number mod 323
108 and 288 are both divisible by 36
Last fiddled with by enzocreti on 20200818 at 21:00
Reason: notice that lcm(215,344,559)=22360 22360/(18*18)=69.01234567... curious I notice also that 541456 (multiple of 43),is  21
