Numbers 420,000s

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420,744

[Math.]   123,448,227,904 = (123,448+227,904)2 = 351,3522  and

420,744,227,904 = (420,744+227,904)2 = 648,6482 and 351352+648648 = 1,000,000.

 

422,481

[Math.]    422,4814 = 414,5604+217,5194+95,8004. This is the smallest integral solution (found by Roger Frye, 1988) to the equation a4+b4+c4 = d4 which was proved by Noam Elkies to have infinite solutions.

 

426,389

[Math.]   A prime number.  One of Markov numbers are… 75025, 96557, 135137, 195025, 196418, 294685, 426389, 499393, 514229, 646018, 925765…

 

428,571

[Math.]   The 6-digit period of the rational number 1/7 = 0.142857142857… or 0.142857. Its multiple by 7 is 999,999.  Then,             142,857×n = abc,def,ghi,jkl and

abc,def+ghi,jkl = 142,857, if n  = 7t+1

abc,def+ghi,jkl = 571,428, if n  = 7t+4

abc,def+ghi,jkl = 285,714, if n  = 7t+2

abc,def+ghi,jkl = 714,285, if n  = 7t+5

abc,def+ghi,jkl = 428,571, if n  = 7t+3

abc,def+ghi,jkl = 857,142, if n  = 7t+6,

for t = 0 to 142,856, and in fact abc,edf = t. The results are digit-rotations of 142,857 and the sums of first and last 3-digit numbers are always 999.