Is this number transcendental?

Last week I was thinking about numbers while stuck in traffic, and I thought of this number:
x = sum 2^-p_k over k = 1, 2, 3, … and where p_k is the k-th prime number,
which is approximately 0.0110101000101000101000100000101… in base two and 0.41468250985111… in base ten.

Based on the fact that it has a non-repeating binary expansion, I know it’s irrational. But is it also transcendental?

I googled around and found Liouville numbers. I think this number is a Liouville number and would therefore be transcendental? Am I correct?

Given the fact that I thought of this number in a spare moment, surely some real mathematician has already documented it. Is there a proper name for my number? (Otherwise, I call dibs!)

Wikipedia: "In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x2 − 2 = 0. Another irrational number that is not transcendental is the golden ratio, φ {\displaystyle \varphi } \varphi or ϕ {\displaystyle \phi } \phi , since it is a solution of the polynomial equation x2 − x − 1 = 0. "

Just seeing the thread title made me guess that this was going to be about a kin of Liouville’s constant.

Looking around and comparing the density of ones in algebraic numbers and this one suggests it is transcendental. But a paper here (PDF) on this doesn’t seem to provide any headway. (Search for “prime” in that paper.)

Googling a bit, I find a pdf on the Fine-Structure Constant which shows an approximate relationship between the Kepler–Bouwkamp constant and Pleonast’s Number. :eek:

Thanks for the references. That let me find that it’s sometimes called the prime constant. I’ll try further searches on that.

It’s also on the Wikipedia list of numbers suspected to be transcendental.

To be honest, I’m a little surprised that this isn’t proven to be transcendental. It looks to be exactly the sort of thing for which such a proof would be easy (you know, for the sort of person who does such proofs, not easy for me).

Intuitively it seems very likely to be transcendental. But it doesn’t appear to me to be a “Liouville number” — aren’t these rare?

There’s a wonderful book with a name like “Sixty Math problems with elementary solutions.”* It contains a proof that e is transcendental, allegedly much simpler than Hermite’s original proof. My eyes glazed over trying to follow the proof, but I vowed to persevere. At some point I misplaced the book and it hasn’t turned up since — fell behind the bedstand? I wonder if losing the book and its eye-glazing proof was Freudian. :smack: Nevertheless I’ll spring for another copy if/when I make it to a good bookstore.

    • I Googled for the exact title and author. But despite that the book must be somewhat famous, Google was no help. Has anyone else noticed that, quite suddenly, Google’s searches are far far worse than before? I Google for simple words or places and get advertisers instead: Google prefers even a misspelled advertiser over what I want.

And just now, preparing for tonight’s bridge session I Googled “contract bridge two club opening step responses” to refresh my bidding system memory. Would you believe I got ZERO relevant responses in any of the first several pages of results?

Base on the list helpfully linked by Asympotically fat (I can’t believe my search* missed that page), it’s definitely not a Liouville number, since if it were, it’d be a known transcendental number.

*Or, maybe I can, if Google is not doing so well.

It’s easy, just check the list of all the algebraic numbers and see if it’s on it. There’s only countably many, so it won’t take too long. :wink:

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If I may hijack mine own thread, I know that the sum of two rational numbers is also rational. Is the sum of a rational number and a transcendental number always transcendental?

Yes, Suppose that x is rational, y is transcendental, and z = x+y is algebraic. Then y = z-x, i.e., a rational number minus an algebraic number. Since a rational number is algebraic, and algebraic numbers are closed under addition, y would be algebraic, i.e., not transcendental.

  1. Liouville numbers are not rare. From the Wikipedia article, there are uncountably many, equal in size to the reals.

  2. OTOH the Lebesgue measure is zero.

What do mean by rare? :wink:

  1. :slight_smile:
    Just “looking” at the definition of Pleonast’s number, which keeps adding on chunks largish compared with any epsilon, it seemed “obviously” non-Liouville. No, I don’t have a proof. :slight_smile:


Am I missing something? It’s definitely not known to be Liouville. But that’s not the same as: definitely known to be non-Liouville.

Thanks, Giles; I could’ve worked that out if I’d thought about it longer, but sometimes it’s easier to type than think.

No, I’m missing a “known” in front of “Liouville”. As in my number is definitely not a known Liouville number.

Originally, I was thinking that primes are sparse enough, but I have doubts and I have no intuition at all here.