There are many approaches you could take to determine when a decimal will equal a whole number. I am not sure that my previous explanation of comparing a circle of circumference of the decimal number to a proportion of a circle of circumference 1 solves anything. The remainder would have to be multiplied to it hits an even number.

Not so easy to picture, but a diagram would show it is just like the Mod of integers. That is a circle with a remainder. However decimals tend to be something we could work with. I know it is cumbersome, but I theorize it would give a better picture than brute force factoring. I am just intuitively making theories on this. It would take a lot to crunch numbers about it.

But if it worked the only thing that would matter on a hundred digit number is the remainder. And perhaps I could research how to use calculus and differential equations to determine when the proper decimal is being approached. (I say research because I only had 2 semesters calculus and this is a new technique.) But you are right that the complexity is a problem, but like all good solutions, intuitively, I say it can be simplified.

The definition of Prime numbers is based on integers being multiplied together. If you were able to use decimals it would break the definition and be meaningless. Or would it since we are determining where those decimals are becoming whole numbers? And if that decimal was determined to become a whole number after the desired value you would know that decimal does not divide into the given number, instead it turns into a whole number, if ever, at a larger number.

Also there is a problem of their being infinitely many decimals. I don’t think you would need to test them all. If your decimal was getting infinitely smaller with many infinite steps to multiply it to a whole number, I don’t think it would be a candidate. Again we’d be using calculus and determining the limit.

I’m sorry my reply was slow. I am just trying to think what makes sense and what is junk. Thank you for the reply. Please let me know if this reply makes any sense. My answer to your comment is that not all decimals need to be used and I leave you with a question: Can all integers be written as decimals rotated (multiplied) by and integer? And the answer is yes. So the real question is: “Is there a useful pattern? I don’t know.

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I wrote the above on Jan 15th and did not post it because it is confusing. What I am trying to do is look for patterns in decimals; specifically between 0 and 1. Yes I know that every Prime number has a decimal remainder when you divide it by an integer, but this is different, I believe. I agree there is mass confusion and a daunting task to program this in a program like Mathematica. I do not have a solution or even know who to program it yet. But if you program it like a modulus (the circle revolutions proportions I talked about) then there is a starting point. The geometry will solve complex series and data. But I realize that this post is confusing and does not yet present a solution.