A representation using reciprocal primes

In summary: I have no idea.In summary, Robert has found a way to represent all real numbers with a subset of the open interval (0,1) using binary decimals. This notation uses the reciprocals of primes, which diverge slowly, to converge to any number in between. This representation is called a "well formed number" and is unique. Robert believes that all numbers in this notation are transcendental and there are no repeating decimals. However, he is unsure if there are numbers that end in ..\overline{01} or ..\overline{10}.
  • #1
RobertCairone
Gold Member
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Is this interesting?

I have a way to represent all of the real numbers with a subset of the open interval (0,1). I write as a binary decimal x = .a[itex]_{0}[/itex]a[itex]_{1}[/itex]a[itex]_{2}[/itex]a[itex]_{3}[/itex]... where x = [itex]\sum -1^{a_{0}+1}/p_{i}[/itex] where p[itex]_{i}[/itex] is the i[itex]^{}th[/itex] prime.

Since the reciprocals of the primes diverges, in this notation, .[itex]\overline{0}[/itex] diverges to negative infinity and .[itex]\overline{1}[/itex] diverges to positive infinity. By changing the digits, this forms a conditionally convergent series that can converge to any number in between. For example, √2 = .11111111010011010011001... We could pick [itex]\pi[/itex], but as the series diverges very slowly that would lead to a long string of ones before the first zero. Still, in principal any number can be achieved.

There are an infinite number of ways to converge to any number, but some are more efficient than others. If we impose a simple rule that if the partial sum is less than the targeted value, the next a[itex]_{i}[/itex] is 1, and if the partial sum exceeds the targeted value then the next a[itex]_{i}[/itex] is zero. If perchance the value is exactly equal, the next digit is a one. This representation is unique, and I call it a well formed number. A test can show if any partial sum steps outside the bounds of a well formed envelope, that is, after a finite number of steps any sequence can be shown to be ill formed, or to convergent to a decreasing range of possible values. The examples here should be well formed.

In this notation, since every prime must contribute something to the value, either as a positive or negative element, all numbers are transcendental. There are no numbers that end in ..[itex]\overline{0}[/itex] or ..[itex]\overline{1}[/itex] I don't think any numbers can be repeating decimals, but I'm not certain of that. A rational number like 5/6 does not have the representation .11, but .110110101010110..., dancing around the value as it converges once again.

In a properly formed number, there can be only so many consecutive digits, and once the value is approached closely enough, I think at most two consecutive digits can appear before it flips. But every so often there must be at least one such a double, as a number that terminated in the repeating decimal ..[itex]\overline{01}[/itex] or ..[itex]\overline{10}[/itex] must diverge. I don't know if I could say the same for ..[itex]\overline{011}[/itex] or ..[itex]\overline{01011}[/itex] but I suspect it is true

Anyway, that's the general idea if I've expressed it well enough.
 
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  • #2
Hi Robert,

My first question about this is: What is the significance of prime numbers here? Could you just as well use any increasing sequence of natural numbers for which the sum of the reciprocals diverges?
 
  • #3
Yes, the natural numbers would work as well, but I chose the primes as a smaller set of basis elements. I was looking for some kind of "minimalist" way of approximating the reals. The slowness by which the reciprocals of the primes diverges made them seem natural. Does anything diverge more slowly?
 
  • #4
RobertCairone said:
Does anything diverge more slowly?
Treating this as a question about slowly diverging series of reals, and reinterpreting as integrals, the following all diverge, steadily more slowly:
∫1/x, ∫1/(x ln(x)), ∫1/(x ln(x) ln(ln(x))), ...
The primes, of course, would correspond to the second of those.
 
  • #5
haruspex said:
∫1/x, ∫1/(x ln(x)), ∫1/(x ln(x) ln(ln(x))), ...
The primes, of course, would correspond to the second of those.

So if the first corresponds to the integers, and the second to the primes, is there a naive interpretation for the third of these? Does this say there is a definable subset of the primes that still diverges?
 
  • #6
RobertCairone said:
So if the first corresponds to the integers, and the second to the primes, is there a naive interpretation for the third of these? Does this say there is a definable subset of the primes that still diverges?
That's a tantalizing question.
 

Related to A representation using reciprocal primes

1. What is a representation using reciprocal primes?

A representation using reciprocal primes is a way to express a number as a sum of fractions, where the denominators are all prime numbers and the numerators are their corresponding reciprocals. For example, the number 12 can be represented as 1/2 + 1/3 + 1/4 + 1/6, using the prime numbers 2, 3, and 2x3=6.

2. How is a representation using reciprocal primes useful?

A representation using reciprocal primes can be useful in number theory and cryptography, as it allows for unique representations of numbers and can help identify patterns and relationships between numbers. It can also be used in algorithms and programming to optimize calculations and simplify complex equations.

3. Can any number be represented using reciprocal primes?

Yes, any positive integer can be represented using reciprocal primes. However, some numbers may have multiple representations, while others may have no representation at all. For example, the number 4 can be represented as 1/2 + 1/4 or 1/3 + 1/6, but the number 5 has no representation using only prime denominators.

4. Are there any limitations to a representation using reciprocal primes?

One limitation is that it only applies to positive integers, as negative numbers and fractions cannot be represented using reciprocal primes. Additionally, there may be cases where a number has a representation using prime denominators but it is not the most optimal or efficient representation.

5. How is a representation using reciprocal primes related to other number representations?

A representation using reciprocal primes is related to other representations, such as prime factorization and Egyptian fractions. It is also a form of a continued fraction, where the denominators are all prime numbers. However, it is unique in its use of only prime denominators and the corresponding reciprocals as numerators.

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