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 2020-06-10, 17:23 #1 enzocreti   Mar 2018 22×7×19 Posts An algebraic trick (456+559*2) *344=541456. 541456 is congruent to 344 mod 559 and pg(541456) is prime!
 2020-06-10, 18:18 #2 enzocreti   Mar 2018 53210 Posts Curious fact (559*5+456) *344=1118344 1118=559*2 And 1118344 is the concatenation in base ten of 1118 and 344 The prime 3251 has this property: 3251*86=279586...2795 is a multiple of 559 and 279586 is the concatenation in base ten of 2795 and 86 3251*172=559172...559 is a multiple of 559 and 559172 is the concatenation in base ten of 559 and 172. 3251*344=1118344...1118 is a multiple of 559... The prime 3251 is the smallest prime congruent to 456 mod 559 541456=(456+559*2)*(559-215) 541456=(3251-3*559)*(559-215) 541456=(215*15-127*13)*(559-215) 541456=43*2^3*[5^2*(2^7+1)-13*(2^7-1)] 541456=2*(4+4*7+4*7^2+559)*(7^3+1) 541456=(3*4+3*4*7+3*4*49+103) *2*(7^3+1) 541456 has also the curious representation 541456=(456+559*2)*(456+559*2-123*10) where in the second parenthesis there are all the digits 1,2,3,4,5,6 so 215, 69660, 92020, 541456 are congruent to plus or minus (456+559*2-123*10) (mod 559) 215,69660,92020,541456 are either multiple of (456+559-123*10)=215 or multiple of (456+559*2-123*10)=344 There are two primes pg(56238) and pg(75894) 56238 and 75894 are multiple of 26 56238 and 75794 are congruent to 86k mod 103 103=559-456 (1230-344*2-1)*1000+1230-559*2+344=541456 1456 seems to return ... pg(56238) and pg(75894) are probable primes If I am not wrong 56238 and 75894 are the only exponents found leading to a prime which are multiple of 26 Both 56238 and 75894 are congruent to 1456 mod 182 56238 is congruent to 91*10 mod 1456 75894 is congruent to 91*2 mod 1456 541456=(645+559*2) (645+559*2-1456)+215 Where 645 is a permutation of 456 The second term in the parentesis is 307, the first 1763 215*10 and 541456*10 are both 1 mod 307 ((x+y) *(x+y-1456)+215)=541456 Wolphram solutions y=1763-x y=-x-307 69660 is multiple of 43 and pg(69660) is prime 69660=645*(215+1)/2 pg(69660) pg(75894) pg(56238) are primes 69660, 75894 and 56238 are either divisible by 645 or 546. Where 645 is just a rearrangement of 546 swapping a digit. 56238=(546+10^3-1) *546/15 So 56238 has the curious representation (545454+546^2)/15 75894 and 56238 have the same residue 24 mod 54 and they are both divisible by 546 69660 is 0 mod 54 and it is divisible by 645 The prime 56239 so has the curious representation (546^2+15+545454) /15 Last fiddled with by enzocreti on 2020-06-13 at 21:14

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