Thread: 69660 and 92020 View Single Post
 2020-03-30, 23:06 #8 enzocreti   Mar 2018 17×31 Posts ... I note also... 69660 I note also that (lcm(215,344,559))^2=4999*10^5+69660-60 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*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) 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 Pg(331259) is prime and pg(92020) is prime. 92020+(92020/215-1)*559+546=331259 Again magic numbers 559 and 546 strike! So 331259 is a number of the form 215*(13s-1)+(13*b-2)*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*8-546)*3-11# 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*18-1) and 69660 (multiple of 43) is 215 mod (18*18-1) 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*18-1)*215 92020=69660+18*18*69+4 -215 is congruent to 108 mod (18*18-1=323) 541456 is congruent to 108 mod (18*18-1) 92020 is congruent to (17*17-1) mod (18*18-1) 69660 is congruent to 215 mod (18*18-1) 108=6^3-18^2 17*17-1+36-216=17*17-1+6^2-6^3=108 215=6^3-1 To make it easier 215, 69660, 541456 are congruent to plus or minus 215 mod 323 92020 is congruent to (17*17-1) mod (18*18-1) 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 6^3-1 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 (17^2-1) mod 323 and to - (6^2-1) 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)+6^3-1=(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 17^2-1 215, 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)^2-1 mod 323 288 is 17^2-1 288 in base 16 is 120 120=11^2-1 also 323=18^2-1 in base 16 is 143=12^2-1 344*((1444456-1763*2^9) /344-1)=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 3-smooth number mod 323 108 and 288 are both divisible by 36 Last fiddled with by enzocreti on 2020-08-18 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