Find a Square

[davar55]

Find a positive integral square whose (decimal) digit representation
ends in the pattern ABABABABAB.

Is it possible to have a square end in a longer repeated pattern?

[Wacky]

For A=B=0, it is trivial.

[grandpascorpion]

Davar, I assume A and B have to differ. Right?

[davar55]

Yes, the number must end in ABABABABAB with A and B distinct.

There is more than one solution, so perhaps
find the smallest.

[grandpascorpion]

Found with the help of Excel, probably not the smallest. My solution is around 4.4 billion and change

4491146011^2 = 20170392492121212121


Yeah, there's an infinite number of solutions. a*10^10 + (any solution under 10^10) where a is an integer >= 1 will give you a different solution.

[davar55]

Perfectly good solution, though not the smallest.

And yes there are an infinite number of solutions.
Another that you didn't mention is four times yours
(double the square root)
since that square ends in 8484848484.

[grandpascorpion]

Quite true

[m_f_h]

162459327
found in 9 min by PARI script (KISS principle, could be improved in roughly as many ways as there are characters in...)
{zz=10^8;for(x=2*10^6,10^10,if((tt=divrem(x^2%zz,10000))[1]!=tt[2],next);
if(!tt|(tt=divrem(tt[1],100))[1]!=tt[2],next);
print([x,tt[2]=x^2%10^10]);if(tt[1]==tt[2]\zz,break))}

[m_f_h]

If x^2 = x^2\10^4 (mod 10^4) and x^2 = x^2\100 (mod 100),
then x^2 = 0,64,21,84,69,29 or 61 mod 100.
(where \ means truncated integer division)

The proof of this theorem exists, even if I don't know it since I don't look for it...

(and anyway, one cannot write into the margins of this forum...)

[davar55]

Yes I think that's the smallest!

My source gave

508853989[sup]2[/sup] = 258932382121212121

which is larger!

Since your solution

162459327[sup]2[/sup] = 26,393,032,929,292,929

ends in 2929292929, we now have solutions that end in
ABABABABAB with AB = 21, 29, and 84.

According to my source, there are two more possible ending values of AB besides 00.
(You listed three: 61,64,69.)

Can you find solutions for these values too?
(I don't have their smallest solutions at hand.)

[m_f_h]

Originally Posted by davar55

Can you find solutions for these values too?
(I don't have their smallest solutions at hand.)

Everybody can, it's sufficient to paste my pari script into gp and change the starting value to (least solution)+1

PS:
1/ hey guys, I have 2 or 3 other things (less funny but more required) to do...
2/ note my Thm speaks about mod 10^8 not mod 10^10 but adding the same thing with a sufficiently large multiple of 100...001 should do the job, no ?
3/ you should optimize my script, maybe by using Mod(,10^10) and adding 2n+1 to go to the next square, instead of doing x^2%10^8 each time.
(check if it's really faster, I had some counter-intuitive surprises regarding similar pbs...)
I don't have time for more than this quick hack, today :-( !
(besides the fact that my box has 2 mprimes running and a dozen of other active windows with a dozen of tabs in each, including gmail with its CPU intensive scripts...