Edgard Neuman Posted October 21, 2019 Posted October 21, 2019 (edited) Hi, Here is a math question : First I'm going to define some things (some names may already exists that I don't know of, so please take my definition into consideration) - let's call p[n] the nth-rank prime number p[0]=1, p[1]=2, p[2]=3, p[3]=5 etc - as you know, each integer >0 can be written as a product of integer powers of prime numbers.. let's call it the "prime writing" of a number... i'll write u[n] so for any integer X we have X = product( p[n] ^ u[n] ) - we can extend this to rational numbers, simply by allowing u[n] <0 My question is : can we define a set of irrational numbers in ]0 ; 1[ that extends p[n] when n<0 and are the building blocks for irrational numbers ? Let's call them subprimes.. Those numbers would have the properties following : - they are not power/products of primes and other sub-primes and of course integer powers of some other real number (other than themselves) Are they already known ? Do they exist ? How to construct them ? I have some (very faint) clue : When you elevate these numbers to positive powers , you get closer and closer to 0.. so the more you go close to 0, the more likely to find a power of a bigger subprime.. so the density must decrease closer to 0.. you get some sort of sieve, but closer and closer to 0. Edited October 21, 2019 by Edgard Neuman
Edgard Neuman Posted October 21, 2019 Author Posted October 21, 2019 (edited) I realize maybe we would have to define them each as a unique "set" of integer powers of a specific irrational number between ]0;1[ but the idea remains the same Edited October 21, 2019 by Edgard Neuman
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