1 13th April 00:31 ioannis External User   Posts: 1 Transcendentals In 1937 Mahler proved that Champernowne's number was transcendental. In fact he proved the much stronger result that if p(x) is a polynomial with integer coefficients then the real number obtained by concatenating the integers p(1), p(2), p(3), ... to get 0.p(1)p(2)p(3)... is transcendental. Champernowne's number is the special case where p(x) = x. Does anyone know if there's any reason why the stronger result with a(n) being a (non-trivial) sequence was not proved? For example, if a(n)=floor[exp(n)], I would expect the number: 0.a(1)a(2)a(3)... to also be transcendental. I can see it, but I cannot prove it. A polynomial p(x) with integer coefficients is just a special sequence. Any references or pointers towards that direction are appreciated. Many thanks, -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/

 2 13th April 00:31 g. a. edgar External User   Posts: 1 Transcendentals Can you also "see" this one: floor(10^n/3) ?? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/
 3 13th April 00:31 ioannis External User   Posts: 1 Transcendentals I believe I said "non-trivial" sequences (which in this context is taken to mean sequences with terms with no repeating patterns). Otherwise feel free to speculate about a(n)=floor[exp(n)]. -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/ [ Moderator's note: "Non-trivial" needs to be replaced by a precise definition in order to get something that might be provable. "Not having a pattern that I can see" will not work. As for the original question, the obvious answer is that if Mahler had known how to prove a stronger result he would have done so. -RI ]
 4 13th April 00:31 ol3 External User   Posts: 1 Transcendentals Maybe the base of the exponential function must be other than any power of the base used for representation? The moral of the story: You need to look at proofs for specific kinds of functions in order to avoid pitfalls. --OL
 5 13th April 00:31 bo198214 External User   Posts: 1 Transcendentals I (and probably you too) can not see a repeating pattern in the sequence of decimal figures of sqrt(2), despite it is not transcendental.
 6 13th April 00:32 ioannis External User   Posts: 1 Transcendentals Probably true, but my question is concerned with a more general construction. For example, if a(n)=floor(sqrt(2)*10^(n-1)), then it appears to me as though 0.a(1)a(2)a(3)... is transcendental. -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/ [ Moderator's note: The point is that in most cases you can't tell whether a number is transcendental by looking at its digits. It's almost certainly true that the number Ioannis mentions here is transcendental, but there's no proof. Conjectures of this sort are very easy to make, but very hard to prove. Unless somebody has something really new to say, I'm closing this thread. -RI]
 7 13th April 00:32 ioannis External User   Posts: 1 Transcendentals Before Robert closes the thread, let me make my conjecture as best as I can: Call a "morphism" a pattern of digits a(n) which is in one to one correspondence with the naturals and further it has the property a(n)+a(m)=a(n+m). For example, the decimal patterns of C_2 (Champernowne's base 2 pattern) are a morphism, because they can be mapped directly onto the Naturals and they satisfy a(n)+a(m)=a(n+m). So are the patterns of C_b, where b is any base in {2,..10}. Simply associate the number 1 with its base b pattern, 2 with its base b pattern, and so on. Although a general non-trivial sequence gives a one to one correspondence between N and its terms {a(n)}, n\in N, the pattern it generates (0.a(1)a(2)a(3)...) is not necessarilly a morphism. Take a(n)=floor(exp(n)), for example: a(1)+a(2)=2+7 =/= a(3)=20. The case I have and I am interested in is the sequence a(n) described in this webpage: http://ioannis.virtualcomposer2000.com/math/Naturals.html Although it looks that this a(n) is not a morphism (a(1)+a(2)=102+101022=/=101021010222=a(3)), on another level it looks like *it is* a morphism, since the word for a(n) is essentially nothing but a string representation of all the preceding naturals, 012...n-1, therefore a(n) is basically a different representation of n itself. On some level then (which sorry, I cannot put to words) c(n) is essentially Champernowne's expansion, expressed using different symbols. Therefore it looks as if A is transcendental iff C_{10} is. I am sorry, this is the best I can do to describe the problem. I seem to be lacking the description tools in this case. -- I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/