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)*(559215)
541456=(32513*559)*(559215)
541456=(215*15127*13)*(559215)
541456=43*2^3*[5^2*(2^7+1)13*(2^71)]
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*2123*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*2123*10) (mod 559)
215,69660,92020,541456 are either multiple of (456+559123*10)=215 or multiple of (456+559*2123*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=559456
(1230344*21)*1000+1230559*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*21456)+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+y1456)+215)=541456
Wolphram solutions
y=1763x
y=x307
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+10^31) *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 20200613 at 21:14
