Complicated Number Theory problem

[PiIsGreat]

Hey guys,
we were asked by our profs to ask challenging problems in the lectures. One was asked by a student that nobody really seemed to get at first sight and it looks very complicated if you solve it.
It goes as follows:
Given is a real number a in its decimal representation, where each decimal digit is a prime number. The decimal digits are arranged along a path as seen in my diagramm. For each m ≥ 1, the decimal representation of a real number r_m is formed by writing before the decimal point the digit 0 and after the decimal point the sequence of digits of the m-th line from the top read from left to right from the adjacent arrangement. In an analogous way for all n ≥ 1 the real numbers c_n with the digits of the n-th column to be read from top to bottom. digits of the nth column from the left.
How they are arranged:

The task is to show that every c and and r is rational if a is rational. Me and my fellows found that here is the sequence to be found online: 1,2,9,10,25,26,49,50 - OEIS
We don't yet have a specific idea although one of my friends recommended looking at every second odd index from the rows. Have you got an idea on how to proof it with me together?

 

[Steven G]

I myself would not give you a hint as you need to suffer through these type problem. Solving problems like this on your own-or in a group-is really beneficial for you.
Please spend some more time. Post your results and then helpers here will be more willing to give you a hint.

 

[PiIsGreat]

I myself would not give you a hint as you need to suffer through these type problem. Solving problems like this on your own-or in a group-is really beneficial for you.
Please spend some more time. Post your results and then helpers here will be more willing to give you a hint.

I have spend more than a day just thinking about it. I also found some properties about the lines such as -> the first row can be rewritten as a term (you can find that online too) but its very hard for me to wrap my head around that problem. I just can't connect the periodical stuff with the properties of the listing of decimals. My problem is that im missing the first spark, like: How do I approach that? Would i do it with a=p/q=a1/10+a2/100+a3/1000...
Should I do it considering the rows? Why does it matter if the decimals are primes? Does this make the problem solveable or only easier? For me, it just seems as a non helping statement.
I can assure you that I tried many things but the problem is that i can't get a starting point and thus can't produce more outcome to give you and all the other admirable helpers.

 

[Steven G]

Spending a day on a problem is not very long. I spent up to four days thinking about how to solve a problem. Just stick with it and something will eventually click.
Post some of your work

 

[PiIsGreat]

Spending a day on a problem is not very long. I spent up to four days thinking about how to solve a problem. Just stick with it and something will eventually click.
Post some of your work

I know that it might not be long but it is definitly long if you dont have a starting idea. I started examining the first row and noticed various things.
1. The numbers are always either squares of odd numbers or their followers. Thus you can find the decimals in that row by using f(x)=4x²+4x+1 and f(x)+1.
2. I noticed that from a certain point onwards, every row follows this closely, so that in every row from a certain point onwards you get the odd decimals via f(x)-r and the even ones via f(x)+r
3. I noticed that i can argue almost the same (but with a different function) for the columns.
4. I noticed that the difference between the selected decimals in each row strictly increases which might be helping but idk yet.
5. I noticed that i still dont get, why there is that exlusion of non primes.

 

[PiIsGreat]

I meant f(x)+1-r.

 

[PiIsGreat]

Spending a day on a problem is not very long. I spent up to four days thinking about how to solve a problem. Just stick with it and something will eventually click.
Post some of your work

I've now gotten to a point where in only need to proof that the rows are always periodical after double the length of the period of alphas decimals. How can I proof that? Im dead stuck, ive noticed that the periodical thing always starts after the main diagonal of the grid and that you might want to reduce it modulo 2p but i cant get that last step with the periodical thing after 2p (periodical lenght of alphas decimals). Have you got a last advice?