I would like to know if someone knows how to prove that there are no perfect square composed only by zeros and ones in their decimal representation whose prime factors are only 3 and 7 (so of the form $N=3^{2a}\cdot 7^{2b}$). (For example, numbers like 10101001, 1001010100001,...)
using Mathematica with this code:
Select[Range[1, 10000000], Complement[IntegerDigits[#^2], {1, 0}] == {} &]
I have verified that there are no such perfect square numbers below $10^{14}$
In my particular case I know that the only prime numbers which divides this N are 3 and 7. So we know that N is a perfect square composed only by zeros and ones and it's of the form $3^{2a}\cdot 7^{2b}$ for some $a,b\in\mathbb{N}$. Going further I note that if such N exists, then it's of the form $N=9^{10t}\cdot 49^{2r}$ or $N=9^{10t+5}\cdot 49^{2r+1}$for some $a,b\in\mathbb{N}$.
I would like to show that such N can not exists.