[enzocreti]

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