A congruence argument mod 11 settles the other cases in the negative. Please do not share this thread with others; I’d like to propose the problem for the USAMO. Jim On Tue, Jun 30, 2020 at 11:13 PM James Propp <jamespropp@gmail.com> wrote:
Good point!
Jim
On Tue, Jun 30, 2020 at 11:09 PM <rcs@xmission.com> wrote:
It looks like your first example with 25 digits generalizes, just by inserting more 0s, to any odd^2 size. --Rich
----------- Quoting James Propp <jamespropp@gmail.com>:
The title of Eric's post gave me an idea for a different game to play with numbers.
The prime p=101030101 could be called a "square prime" (in base ten) because the number of decimal digits of p is a perfect square AND if you write the digits of p in the form of a perfect square you get a pattern with all the symmetries of a square:
1 0 1 0 3 0 1 0 1
There are 44 square primes with 9 digits:
101030101, 101060101, 111181111, 111191111, 121272121, 121282121, 131333131, 141484141, 141494141, 151545151, 151585151, 161636161, 171767171, 181888181, 191939191, 191969191, 303050303, 313111313, 323222323, 323232323, 363646363, 363676363, 363686363, 373717373, 383828383, 383838383, 383888383, 727212727, 727272727, 727282727, 757545757, 757585757, 777767777, 797939797, 797949797, 909070909, 919141919, 919171919, 919191919, 929292929, 959555959, 979797979, 989868989, 989898989
On the other hand, there are only two 9-digit "square squares": 121242121 = 11011^2 and 404090404 = 20102^2.
There are no 16-digit square squares (probably for congruential reasons but I don't see what they are; do you?).
The are six 25-digit square squares: 1020100000204020000010201 = 1010000000101^2, 1020102020205020202010201 = 1010001000101^2, 1020104040208020404010201 = 1010002000101^2, 1232120202329232020212321 = 1110009100111^2, 1232122422348432242212321 = 1110010100111^2, and 1232124642369632464212321 = 1110011100111^2.
(Note the one whose square root is not a palindrome.)
Are there infinitely many square squares?
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