This is an excellent and notoriously difficult question. The short answer is:

This is a famous open problem. There is no known proof that (or any for ) is transcendental. It is, however, widely conjectured to be true.

Let's break down why this is so hard and what is known in this area.

1. The General Conjecture (Borel's Conjecture)

Your question is a specific instance of a more general set of ideas and conjectures, often attributed to Émile Borel (around 1903, with more formal statements later, e.g., in his 1950 paper). The conjecture, in a modern form, states:

Conjecture: Let be an irrational algebraic number. Let its base- expansion be . Let be an integer base such that and are multiplicatively independent (i.e., not both powers of the same integer). Then the number is transcendental.

In your case:

• (irrational algebraic).
• The base is .
• The new base is .
• Since and are multiplicatively independent, the conjecture predicts that your number is transcendental.

This problem is considered to be at the absolute frontier of transcendental number theory, deeply connected to the unresolved mystery of the statistical properties of the digits of algebraic numbers.

2. Why is this so difficult?

The difficulty lies in our profound ignorance about the sequence of digits of algebraic numbers like .

• No Known Structure: We can compute the digits of to arbitrary precision, but we have no "formula" for the -th digit, nor do we know of any pattern or structure it must obey, other than the fact that it is not ultimately periodic (since is irrational).

• The Normality Conjecture: It is conjectured that all algebraic irrational numbers are normal in every integer base. This means that every finite block of digits of length appears with the limiting frequency . For in base 10, this would mean that the digit '7' appears 10% of the time, the block '42' appears 1% of the time, and so on. This conjecture is completely open. If were proven to be normal in base 10, its digits would appear "random".

The problem of proving is transcendental is considered to be of a similar, if not greater, difficulty than proving that is normal.

3. Connection to Modern Transcendence Theory

The most significant modern breakthroughs connecting the properties of a number's expansion to its transcendence come from the work of Adamczewski and Bugeaud, building on the work of others like Loxton, van der Poel, and Cobham.

Their work revolves around the notion of subword complexity (or factor complexity) of a sequence. The complexity function of a sequence counts the number of distinct blocks of length .

• For a periodic sequence (rational number), is bounded by a constant.
• For a "simple" non-periodic sequence, like the Thue-Morse sequence, grows linearly, i.e., .
• For a "random" or normal sequence, the complexity is maximal: for base .

A cornerstone result in this area is a theorem by Adamczewski and Bugeaud (2005):

Theorem (Adamczewski, Bugeaud): Let be a real number. If the subword complexity of its base- expansion satisfies , then is either rational or transcendental.

This is a powerful tool for proving transcendence for numbers with "low complexity" digit expansions (like Sturmian numbers or automatic numbers).

How does this apply to your question?

It doesn't, and that's precisely the problem. The digits of are conjectured to be normal, which would mean its complexity in base 10 is . This is the highest possible complexity. The ratio goes to infinity very quickly.

The existing theorems prove that numbers with very structured (low complexity) digit sequences are transcendental. Your number is constructed from a digit sequence that is believed to be completely unstructured (high complexity). We currently have no methods to prove the transcendence of numbers defined by such "random-looking" digit sequences.

4. What we do know (and what it implies)

• Cobham's Theorem (1969): The sequence of digits of an irrational algebraic number (like ) in any integer base cannot be generated by a finite automaton.
• Loxton and van der Poel's Theorem (1987): A number whose base- expansion is generated by a finite automaton is either rational or transcendental.

These two theorems together imply that if we form a number whose base- digits are generated by an automaton that is not ultimately periodic, then must be transcendental.

Your number is formed from the digits of , which are not automatic. Therefore, these theorems do not apply. This again highlights the divide: we have tools for the "simple" case (automatic sequences) but not for the "complex" case (digits of algebraic numbers).

Summary

Proving that your number is transcendental would be a landmark achievement in number theory. It would require a completely new insight into either:

1. The structure of the decimal digits of , or
2. A new method for proving transcendence that works for numbers with high-complexity digit expansions.

Both of these are considered among the hardest open problems in the field. So, for now, the answer remains: It is conjectured to be transcendental, but a proof is far beyond our current reach.