primes squared inf showing a pattern with the last digits
1. so if i take a prime and square, cube, etc inf times, the last numbers will (so far) be four numbers repeating inf.
2. if i take 5 to the 3rd power and up inf times the last 3 numbers will alternate between 125 and 625.
My background in this is type of working numbers is non existent. Up until about 2 months ago when i started trying to create godel type theorem's because i was bored and couldn't remember (Goodstein's)? theorem i had seen in a book by R. Penrose a few years ago. Then I started squaring numbers to keep myself busy, getting to a billion in almost no time, and doing all the multiplying in my head easily. Thus I started seeing patterns, trying to turn them into functions and theorem's but for #2. For example the function of x was 5 to the 3rd power + n , followed by the sideways M with inf on the top and n + 1 on the bottom, now after = i set s1 and s2 to the 125 for s1 and 625 for s2, with the double arrow between them. I'm sure this is wrong. For #1, I don't believe this anything new but can anyone help me with working things like this properly into theorem's? Only recently have I become so obsessive with numbers, its happened before but i pushed it away and now i find myself doing these things again. I'm 28 and when i was younger i was "ok" with math but wherever this is coming from I could use some help in what to study as well because I find i enjoy working numbers almost obsessively. I just have only done avionic systems in the navy and up to college algebra in college. I'd have to look but i believe i did find a set of propositions and rules, ie the axioms to work 1-3 squared up and up and up to zero. Even if, its reinventing the wheel anyway. I could use any help or suggestions you'll give, good or bad, i just want to self teach myself far enough to get a degree in physics or mathematics. As well as know what I'm doing.
For #1. other then 5 just repeating 5 5 5 5, and this works for 2 but does not use the typical 1 3 7 9 pattern either. otherwise, the last numbers are as follows, they repeat as you go up in powers.These numbers also show up in a list of large primes as the last digits, i did this yesterday in half an hour in a class is there more i can do with this possibly? I would estimate, and thus will check higher up numbers to make sure but besides 2 and 5 i believe this pattern will hold and possibly have yet another pattern to it.
Notice the pattern here as well with 1 3 7 9 also. I would conjecture any prime other then the likes of 5 would keep these same numbers. Ill work on it i may be wrong.
Re: primes squared inf showing a pattern with the last digits
i was wrong i believe now that it may be every other prime has this pattern. thats what i get for spending 30 min on something. The same 4 do show up on every other prime number itself starting with 3. I suppose i got excited for no reason.