N>1 roots of primes are irrational

The question came up in a conversation recently, and although it seemed intuitively true, I couldn't find a proof online. Many of you reading this are probably familar with the proof for the irrationality of √2. Well, this is a more general version for all n integer roots of any prime.

Theorem

Proof

Euclid's first theorem states that for a, b ∈ Z and any prime p:

p|ab  →   p|a  or  p|b 

An immediate corollary of it is, for a, 1 ≤ n ∈ Z and any prime p:

p|aⁿ   →   p|a

Since

p|aⁿ = aa^n-1    →  p|a  or  p|a^n-1

but,

p|a^n-1 = aa^n-2   →   p|a  or  p|a^n-2

:

p|a^n-(n-2) = a² = aa   →   p|a  or  p|a

→  p|a  or  p|a  or  p|a  or... p|a  ( n times )   →   p|a

* * *

Now suppose for contradiction that

∃a,b,2≤n∈ Z, p∈P (p^1/n = a / b), where a nad b have no common factor.

p^1/n = a / b   →   p =  aⁿ / bⁿ 

→   pbⁿ =  aⁿ  →   p|aⁿ   by def. of divides, since bⁿ∈ Z

→  p|a by Corollary to Euclid's first theorem.

Hence a = pk ,  k∈ Z by def. of p|a.        (1)

But  p^1/n = a / b   →   a = p^1/nb = pk   →   pbⁿ = pⁿkⁿ 

→ bⁿ = p( p^n-2kⁿ )  →   p|bⁿ   by def. of divides, since p^n-2kⁿ∈ Z

→ p|b by Corollary to Euclid's first theorem.

Hence b = pm ,  m∈ Z by def. of p|b.     (2)

Statements (1) and (2) jointly contradict the statement that "a and b have no common factor".

Hence ∀a,b,2≤n∈ Z, p∈P (p^1/n ≠ a / b)  q.e.d.

UNSAFE PROFANATION

An artist doesn't need a lecture on the value of art or the creative process in general. It is a blessing of the muses, a charitable gesture of serendipity for which all artists yearn like junkies. Why is that? Because both the creative process and inspiration are what an artist's soul feeds on and is defined by, which makes them sacred. They're sacred gifts whose value largely springs from their ephemeral and inconsistent nature due to the muses' capriciousness.

Therefore it seems natural that to a genuine apprentice of the muses the acquisition of a respect for the value of this gift and the recognition of its instantiations in the form of "perceivable artworks" is inevitable. It follows that the role of an art school should be both to aid the apprentice of the muses to evoke the longing for this gift, and nurture a respect for its instantiations.

It has come to my attention that during a recent social function at the Queensland College of Art (Griffith University), students have been physically tampering with an unfinished sculpture of one of their absent colleagues. This included climbing on the sculpture en masse, and ultimately soiling it with sand and what appears to be alcohol. Whereas such behavior should be generally labeled as disrespectful or thoughtless in the least, within the artistic community it is nothing short of a profanation.

One may be tempted to endorse a more sympathetic stance, and claim that the students deserve a measure of consideration due to their sheer ignorance. As a proof of their naivety serves the fact that by publishing their photos of the function's activities on Facebook, they did not care to exclude those of the sculpture being tampered with. To the contrary, judging by the number of photographs depicting the activity in question, it seems that it was one of the evening's highlights. Imagine the fury of young "Phidias" upon discovering that the profanation of the progeny of his inspiration served as a vehicle of base amusement to a herd of intoxicated morons.

Pleading ignorance, may get the students off the hook, but the story does not end there. A senior  lecturer and a PhD student were present at the function, as evidenced by the published photographs, which makes them passive participants. What is one to say of the ignorance of those senior figures of an artistic and academic institution? Why didn't they react? Did they succumb to a momentary primitive group think? Were they so overwhelmed by the libations as to be completely oblivious of the surrounding goings on? Or maybe their formal statuses are merely empty labels masking equally ignorant individuals? Surely all those alternatives seem equally unthinkable!

The senior academics are denied a plea of ignorance. If that was granted, what would that imply about the school (QCA) which they represent? After all it would be dreadful to permit the thought that the "Art" in QCA is also just an empty label. So let's give the school the benefit of the doubt and suppose there does exist a genuine spirit of art within its walls. But that would seem inconsistent with the described incident!

What went wrong? Consider this. I'm quite sure that once the story surfaces, the highest ranking bureaucrats of Griffith University Inc. who are more concerned with avoiding liability suits, and making a profit then actually focusing on educating the youths would be appalled that such misconduct had occurred: "damaging a piece of art without taking the appropriate safety measures? The photos clearly depict students climbing the sculpture without adequate headgear and goggles! This is an outrage!".

A lesson from Bach

This anecdote has been attributed to the biography of young Johann Sebastian Bach.
One evening, Johann was playing his latest sonata to a German baroness at her estate. When he had finished, the baroness exclaimed with a bewildered delight: “Ah! Maestro! It was wonderful!… But what did you mean by it?" Bach promptly approached the keyboard again, and replayed the sonata. When he had finished, he turned to the baroness and explained: “That, dear madam, is what I meant.”

Tangibility of the transcendental

I was doing a bit of reading about transcendental numbers today, and had an idea of constructing my own well defined transcendental. I was inspired by the Liouville constant.

Anyway, my idea was to construct a number which is a base 10 concatonation of the digits of consecutive prime numbers:
m_p =  0.2357911131719232931...
or the natural numbers:
m_N = 0.12345678910111213...
The construction proceeded as follows. First we define the function lambda:

Now we can express the numbers as infinite sums:

Where p_n is the n'th prime.

Initially I thought this to be an exercise in creative procrastination, since firstly the idea is dead simple, and secondly who would be silly enough in going to the trouble of formulating such numbers. Surely only a fresh graduate with too much time on his hands.

To my surprise the idea behind those numbers has been already formulated by not one, but a few mathematicians. My m_N is in fact a constant previously formulated by D. G. Champernowne. And it turns out that m_p is already known as the
Copeland-Erdos constant:

Notice the striking similarity in the formulation - I have formulated m_p completly oblivious to the Copeland-Edros formulation. The similarity lies in the fact that my lambda function acts a lot like the floor of base 10 logarithm of y, plus 1. In fact the relation is the following:

Which makes the sums in the exponents of the definitions of m_p and the Copeland-Edros constant equal, since for each i we subtract 1 i.e. we subtract 1, k times, we need to add k back to the sum.

Well, I guess I feel better knowing that I'm in good company deeming quite obscure and prima facie pointless ideas valuable. I knew that both m_N and m_p are irrational, but now I also know that m_N is in fact transcendental (mission accomplished!), and hence don't have to provide a proof, which i feel would be a herculean task.

Who is lying?

Here's a rather simple and fun logic puzzle.

We have three people; Alice, Bob and Cecil. One of them is a liar. You have to determine who is the liar, and give reasons for your choice, based on the following information. Alice claims that Bob is a liar. Bob claims that Cecil is a liar. Cecil claims that both Alice and Bob are liers. Who's lying and why?

Answer in comments.

Infinitude of primes, courtesy of Euclid

I love this proof:
Take any finite list of prime numbers p1, p2, ..., pn. It will be shown that some additional prime numbers not in this list exist. Let P be the product of all the prime numbers in the list:
P = p1p2...pn. Let q = P + 1: 1 more than this product. Then, q is either prime or not:
- If q is prime then there is at least one more prime than is listed.
- If q is not prime then some prime factor p divides q.
This factor p is not on our list: if it were, then it would divide P (since P is the product of every number on the list); but as we know, p divides P + 1 = q. Then p would have to divide the difference of the two numbers* which is (P + 1) − P or just 1. But no prime number divides 1 so there would be a contradiction, and therefore p cannot be on the list. This means at least one more prime number exists beyond those in the list. (Wikipedia)

* If some integer k divides two integers a, b then k divides a-b.
This will be immediate to some, but I'll provide a small proof of that fact for clarity's sake.
Proof:
k divides a -->  a = kx, where x is an integer, by def. of "divides"
k divides b -->  b = ky, where y is an integer, by def. of "divides"
a-b = kx - ky = k(x-y)
but x-y is an integer
Hence k divides a-b, by definition of "divides".

Arithmetic sequences in digits of primes

THEOREM
Here's a fun fact about prime numbers: there are only 5 prime numbers, whose digits (there are bound to be more if we consider positive integers) express an arithmnetic sequence with common difference 1. They are 23, 67, 89, 4567 and 23456789.

123456789 is not a prime.

PROOF
By exhaustion :)