Thread: 69660 and 92020 View Single Post 2022-01-03, 13:06 #14 enzocreti   Mar 2018 10000110102 Posts 92233=427x6^3+1 92233=6^3=51456 mod 427 ((92233-51456)-1)/3-12592=10^3 12592 divides 541456 PG(51456) and PG(541456) are primes 92233-51456=1=69660 mod 1699 13592=1699*8 43*(((51456+(3^6-1)*4)/4)-10^3)=541456 69660-(487-51456)=69660-(92233-51456)=0 mod (71x1699) (69660-487)=0 mod 313 (69660-486)=0 mod 427 486=162x3 162 divides 69660 -449449-13=64=541456+13 mod 139 so 541456=51 mod 139 (449x13=-1 mod 139) 92020=541456+13-449449 331259=71*6^6=259 mod 331 331259=44 mod 71 331249-44=331215 that Is the concatenation of 331 and 215 331259-44-215 Is a multiple of 331 44+215=259 71x6^6-331259+92020=6^2-6^5 mod 311 (6^2-6^5)=7740 which divides 69660 (7740=1 mod 71) multyplying by 9 both sides 71x6^6x9-331259x9+92020x9=-69660 mod 311 331215 is the concatenation of 331 and 215 331+215=546 546 divides 75894 and 56238 pg(75894) and pg(56238) are primes 75894-92020/2=215x139-1 546=215+331 215x139 can be cancelled in both sides this leaves 46009-46010=-1 56238 can be factorized as (139-6^2)*(331+215)=56238 probably there is something in 79*3^j pg(79) is prime by the way 79*3^2=711 and 69660=71x711+2131x3^2 79*3^3=2131+1 71x79+2131=7740 which divides 69660 79x3^2=1 mod 71 69660-19179=71x79x3^2 239239=-9 mod 12592 541456=239239+9 mod 12592 92020=3867+9 mod 12592 23004*46009-1-(92020-6)=3539x13x23003 i think that there should be some group in action... pg(75894) and pg(56238) are primes with 75894 and 56238 multiple of 139 75894+56238=132132=1 mod (1861x71) 331259=1 mod 1861 75894+56238=331259 mod (1861x107) 331259-132132=107x1861 75894+56238-1=71x1861 107=71+6^2 56238 and 75894 are congruent to +/- 6 mod 132 in particular (75894+6)/132+1=24^2 so 331259=56238+75894+92020+107x1001 pg(394) is prime 1323=-1 mod 331 pg(1323) is prime 1323=72 mod 139 1323=(1001-72) mod 394 i suspect that this is connected to the fact that 331259=-72 mod (1001x331) 331259=-1323 mod (1001-72=929) 331259=-394=-1323 mod 929=1001-72 3x6^3-3=394x(72)^(-1) mod 1001 In particolar 394x431+3-3x6^3=169169 92020=1323-394=929 mod 1001 541456+13=1323-394-13=916 mod 1001 (92020-1323+1)=0 mod 449 (541456+13-1323+1)=0 mod 449 541456-449449=92020-13 i forgot to see that 92020=43*2132+344 344 is the residue of the multiples of 43 (69660, 541456, 92020) mod 559 43*2132 is infact divisible by 559 69660-(46009*23005-1)/(313x49)=2^3x3^4=3x6^3 92020=-2 mod (313*49) 313x49x3x6^3-1=215^3 23005*(2+46009x8)=1 mod 429^2 23005x(2+46009x8)=-1 mod (359xp) where p is a prime p=23586469...i wonder if it has some special property the logic behind these primes requires tools that are far beyond current knowledge... 331259=259=71x6^6 mod 331 6^k=1 mod 259 the order mod 259 of 6 is 4 ord(6)=4 infact 6^4=1 mod 259 92020=2 mod 331 92020=4 mod 6^4 92020=71+4=75 mod 259 (92020-4)=71x6^4 239239=-77=-75-2 mod 259 239239=-78 mod 449 541456-77x5837+13=92020 curious that 449449-(239239+78)=210132=13*6^2x449 which is congruent to (6^4-2) mod 2131 239239+78 is 449x41x... 449x41 is 1840...anything to do with 429^2-1??? 541456=7740 mod 18404 7740 divides 69660 18404=(429^2-1)/10=449x41-5 pg(3336) is prime 3336=24x139=-1 mod 71 (24+71x9)x139=-1 mod 71 (24+71x9)x139+1-138=92020 (24x71x9)x139+1)/71=6^4+2 so 92020=(6^4+2)x71-138 or equivalently 92020=71x6^4+142-138=71x6^4+4 ((24+71*15)*139+1)/71=2132 so for example 19179=3^2*2131 can be rewritten as 3^2x(((24+71*15)*139-70)/71)=19179 71x6^6=6^4-44 mod (331x4=1324) 331259=44+215 mod 1324 331259=44 mod 71 (71*6^6-6^4+44)/(139*18-1)=1324 may be 44 is not random... 69660=(44^2-1)x6^2 331259=44 mod (311x5x71) 331259^2=1936 mod (311x5x71) 1935 divides 69660 6^5=1 mod (311x5) 6^5-6^2 divides 69660 1-36=35 I suspect that there could be some link to the fact that 541456=51456+700^2 700^2 is divisible by 35 curious that pg(75894) is prime 75894 is multiple of 139 -75894=(2^16+16) mod (359x197) 2^16=-2 mod 331 2^16=1 mod (255x257) i think that something big is happening on some field sqrt(71x215x3+1)=214 92020=sqrt(71x215x3+1)x430 92020=(71*215*6+1)+429 92020=214^2+215^2-1 pg(51456) and pg(6231) are primes with 51456 and 6231 multiple of 67 curiously (51456/67-6231/67)=26^2-1 19179=648 mod 23004 331259=(19179-648)/71 mod 359 331259x71=222 mod 359 so 222=19179-648 mod 359 note that 331331-(331259-(19179-648)/71)=333 ((92020*3-6)*3*4-1)/71=6^6+1 ...3312648...i think that 3x6^3 has something to do with these numbers... 331259/71=4665+44/71=6^6/10-6/10+44/71=6^6/10+7/355=6^10/10+7/(359-4) 331259=44 mod 4665 106x44+44/71+1=331259/71 4665=359*13-2 331259 =71 mod 44 331259=44 mod 71 so 331259 is a number of the form 115+(5^5-1)k this is curious 43*(1+sqrt(9x+1))=9x solution x=215 215*9+1=44^2 i think that you can obtain a continued fraction from that 69660=6^2x43x(1+sqrt(1+43x(1+sqrt(1+43x(1+... curious fact: 69660=19179=3x6^3 mod 639 19179/3=(639)3 and 6393=-1 mod 139 I think that something is in action over some field... (429^2-6)=-1 mod 46009=331x139 (429^2-6)=3 mod 639 from this 429^2=80^2-1=79x3^4=711x9 mod (71x139) 6393+((71*4+1)*139+1)=46009 i think that in Z46009 something is in action as well as in Z23004 71x6^6-541456=-1 mod 359 541456+14=-261=-331259 mod 359 69660=14=-6^6 mod 359 541456+69660=611116=-261=-331259 mod 359 611116=131x4665+1 92020-(541456+14-98)/13/359=359x2^8 -331259=98 mod (359x13) -(541456+69660)=-611116=359-98=261 mod (359x13) pg(1323) is prime pg(39699=13233*3) is prime 13233=18^2 mod 331 69660=215x18^2 so 13233x215=69660=150 mod 331 and 13233x3=39699=-150 mod (359x111) (359=331+18) 69660=3x6^3 mod 71 (69660-3x6^3)/71=-1 mod 139 this suggests me that something is in action over Z139 or maybe Z46009 23005*(2+331x139)-1 is divisible by 11503 69660-642(=3x6^3-6) is divisible by 11503 23005x(2+46009)-1-(69660-642) is a multiple of 23003 2*(23005*(2+46009*72)-1)/46009-2=331272x10 331272-13=331259 23005x(2+46009x72-1) is divisible by 449 541456+13-449x1001=331259 something mysterious is boiling in Z46009 46009x72-72=71x6^6 x^2/(6^6-2-sqrt(2x+1))-x*(sqrt(2x+1)-1)/215=0 has solution x=92020 min (x^2/(6^6-2-sqrt(2x+1))-x*(sqrt(2x+1)-1)/215)=-19394.4=-19179-215.4 this is a parabola from 331259x71=19179-17^2 mod 359 we have 222=19179-17^2 mod 359 so (2^9-1)=19179 mod (359x13) curious that 19179-511+1 is divisible by (2^2-1), (2^3-1) and (2^7-1) 331259=-98=-(512-414) mod 359 19179=414 mod 139 19179=138x139-3 69660=19179=6^6 mod 639 the inverse mod 639 of 2131 is 427 69660x427=9 mod 639 69660x427-9=639x46549 46549 is prime =6^6-107 I think this is not random but connected to the fact that 107 divides 92020 19179=7x71+14 mod (359x13) 6^6=-14 mod (359x13) 71x270+9=7x71+14=(2^9-1) mod (359x13) from here 6^6=7x71-19179 mod (359x13) form here after some steps... 331x72=7x71 mod (359x13) curious that 19179x(7^4+1)=1 mod (359x13) 3x6^3+70 is divisible by 359 (331259-5) is divisible by 717x6=6x(3x6^3+69) 6x(3x6^3+69)=-1 mod (331x13) so (331259-5) is divisible by (331x13-1) (331259-5)/(331*13-1)-(92020-5)/239/77=72 331x13=-5 mod 359 359x12=-1 mod 139 331259 has the curious representation: 65^2*77+77^2+5=331259=325325*77+77^2+5 further steps toward a theory of these numbers need super-tools (331259-5)/7-6^6=666 6^6=-666 mod (239x11) anything to do whit 92020=5 mod (239x11) 331259=5 mod (239x11)??? curious fact: (239239-(6^6+666)+2)/111=1729 the Ramanujan number I will call these primes Neme primes (Neighboured Mersenne) Neme(3)=73 Instead of Neme primes I could call them Hopeless primes, no hope to find a logic behind them curious that 69660=-342^2 mod (432^2) I think that starting from 23005x(2+46009x8)=1 mod 429^2 and 23005x8=-1 mod 429^2 one can develop someting useful Using Chinese remainder theory numbers multiple of 23005 (=0 mod 23005) and =1 mod 331x139 should form a ring or something similar...and I think that in that ring one can get something (2+46009x8+2x92020) is divisible by 92019 92019x6-92020x5 is a multiple of 3539 a wagstaff prime... 331259=2132 mod 3539 6^6-3x6^3-1=3539x13=46007 239239=-13 mod (9202) (239239+13)/107=2236 69660+2236x10=92020 331259=331 mod 2236 239239x2236x2=-(331259-331) mod 331 239239x4472=72 mod 331 2236=6^2x239239^(-1) mod 331 22360=(19^2-1)x239239^(-1) mod 331 from here 22360=-148 mod 331 239239=-(148/2) mod 331 22360=257-74 mod 331 239239=257 mod (331x19^2) (331259=257) mod (71x111) 331259x6^2=(22360-257) mod 71 I have the impression that powers of 6 and 71 are involved in the logic behind these neme primes 92020+257-14 (257-14=243 a power of 3) is 359x257 331259-243 is a multiple of 257 1001-((331259-243)/257-359)=72 69660=13 mod 257 69660=14 mod 359 there is a logic but it is so complex that it is almost hopeless to find a pattern 331259+14+84=71x13x359 541456=84 mod 359 69660=-345=-(331+14) mod 359 6^6=-14 mod 359 i cannot put toghether the entire pieces of the puzzle anyway 331259=6^6-541456 mod (359x13) (69660-6^6+331)*2-14=6^6 92020=10 mod 3067 331259=22 mod 139 331259=23 mod 3067 331259-23-92020+10=239239-13 92022x36-6^3=71x6^6 71x6^6=6^3 mod 3067 71x6^3=-1 mod (313x7^2) 71x6^3=1 mod 3067 I would call these prime Neme primes or maybe desperate primes I stronly suspect that the exponents of these neme primes are connected among them with a logic that it is impossible to understand...only a God could find a pattern...or maybe a new Gauss... curious that 541456=353(7)9 mod (3539x13) with that "7" inserted 92020=6 mod (359x13) in other words 541456=3539x10-10-1 mod (3539x13) curious that (541456-3539x10) is divisible by 23003=71x2^2*3^4-1 71x6^6=72 mod (3539x13) -(541456+13)=787 mod 858 -331259=787 mod 858 69660=162 mod 858 92020=214 mod 858 359x239=1 mod 429 541456=0 mod 787 787-13=774 divides 69660 787-429=359-1 so for example 331259=429x774-787 774 divides 69660 429=sqrt(92020x2+1) -541456=344 mod 774 there is a hidden structure it is clear that 331259-774 is a multiple of 4601 and 4601 divides 92020 -541456-13=456+331 541456+13-456=359x11x137 331259=-358 mod (773x429) 358=359-1 773=774-1 331259+773 is divisible by 1297 a prime of the form 6^s+1 331259=(259-215=44) mod 71 331259=259 mod 331 71x6^6=259 mod 331 there is something... 23005*(2+46009*k)-1=N^2 for k=8 for k=3680 ,... 23005*k+1 is a square k=8 k=3680 ... 3680*23005+1=(3x3067)^2=9201 (71x6^6-216)/3/3067=359+1 92020=10 mod (3067x13) (9201^2-1)/4601/23-13=787 Last fiddled with by enzocreti on 2022-03-06 at 12:35  