An algebraic trick

[enzocreti]

(456+559*2) *344=541456.

541456 is congruent to 344 mod 559 and pg(541456) is prime!

[enzocreti]

(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