## Wednesday, 4 July 2012

### repdigit endings to squares

James Tanton ‏@jamestanton Wrote:

"2^2 ends with 4 and 12^2 ends with 44. Is there a square than ends 444? How about one that ends 4444?"

To which I think the answer is yes,..... and no.
38 squared is 1444
462 squared is 213444
538 squared is 289444 and there are many more... (so I'm pretty firm on the "yes".)

but I don't think you will ever see a square ending in 4444. Here's why:
This is a list of the square numbers that end in 444 from 1 to 100,000,000

00001444
00213444
00289444
00925444
01077444
02137444
02365444
03849444
04153444
06061444
06441444
08773444
09229444
11985444
12517444
15697444
16305444
19909444
20593444
24621444
25381444
29833444
30669444
35545444
36457444
41757444
42745444
48469444
49533444
55681444
56821444
63393444
64609444
71605444
72897444
80317444
81685444
89529444
90973444
99241444

Notice a pattern? Look at that digit in the thousands place...

Those are the squares of these numbers:
0038
0462
0538
0962
1038
1462
1538
1962
2038
2462
2538
2962
3038
3462
3538
3962
4038
4462
4538
4962
5038
5462
5538
5962
6038
6462
6538
6962
7038
7462
7538
7962
8038
8462
8538
8962
9038
9462
9538
9962

and another pattern appears, that may help us with a more formal proof:

I hadn't thought too much about the idea of digit repeats at the end of squares, and maybe you haven't either, so here is a quick question to get you started.
When speaking of square numbers:
What ending digits can repeat 2, 3 or more times? More particularly, is there any ending digit that repeats four (or more) times?