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Old 2022-02-07, 09:15   #10
robert44444uk
 
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Jun 2003
Suva, Fiji

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You have to love the projected result of this exercise - there is exists within the decimal representation of pi, (which is itself just one of an infinite number of irrationals), all of the infinity of primes, in order.

I would expect that this projected result will be the same for all integer bases, i.e. the representations of the primes in any integer base can be found in order within the representation of pi in that base. I can almost visualise this must be true for base 2.

Can we go further to project or hypothesise that the primes, as depicted in any integer base are in order in the representation of any irrational number in the same integer base?

As integers represented in irrational bases appear themselves irrational, then we can safely say that the primes in order, represented in irrational base form, cannot be found in the representation of the irrational in that base, as the irrational in its own base will have a simple non-irrational form.

Because the representation of the irrational form is infinite, I would hypothesise that the primes represented in irrational base form (which I think are irrational representations themselves) are not included in order in the representation of an irrational number presented in an different rational or irrational base.

Have I covered all the permutations here?

PS: My teacher at school did day "don't mess with infinity, it bites"
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