Just playing with square numbers, I made an interesting observation.
$$11\times 11=121$$ $$12\times 12=144$$ $$13\times 13=169$$ $$.$$ $$.$$ $$.$$ $$20\times 20=400$$ $$21\times 21=441$$ $$.$$ $$.$$ $$.$$.
So, we will go from left to right and take two number to form a new number from the square numbers I listed above. For example from the number $169$ we can make three numbers $16,19,69$ but numbers $61,91,96$ are not allowed. Similarly for four digit numbers like $2401$ we are allowed to take $6$ numbers which are $24,20,21,40,41,01$ but we are not allowed to count $42,02,12,04,14,10$.
Now, look at the number $289$. It is square of $17$. We see that we can select three two digit numbers $28,89,29$. Notice that $28$ is composite while $89$ and $29$ are primes.
Next notice the number $256$. It is square of $16$. We can select three two digit numbers $25,26,56$ all of which happen to be composite.
So, now we are in the position to have definitions:
A Mama's number is a number $x>10$ such that all the two digit combinations of numbers chosen from $x^2$ from left to right are composite. Some trivial examples of Mama's numbers are $24$ as $24^2=576$ and $28$ as $28^2=784$.
A Papa's number is a number $y>10$ such that all the two digit combinations of numbers chosen from $y^2$ from left to right are primes.
The numbers which contain a prime as well as a composite in two digit combinations from left to right are neither Mama's nor Papa's numbers.
I checked from $1$ to $50$ by hand and sadly I found no Papa's numbers. While several Mama's numbers were $15,16,18,20,22,24,25,28,30,38,40$.
Now, I have two questions:
- Are there infinitely many Mama's numbers?
- Is there any Papa's number?
I hope I made myself clear. I beg you to point out if there are any errors in calculations or typos (or edit them yourself).