1
vote
3answers
92 views

What is the difference between these two propositions? [duplicate]

My text says: Let Evens be the set of even integers greater than 2, and let Primes be the set of primes. Then we can write Goldbach’s Conjecture in logic notation as follows: $ \forall n \in ...
2
votes
3answers
42 views

Logical structure of elementary number theoretic proof

I'm trying to understand the detailed logic structure of a proof by use of the bezout identity.The number theoretic part i easily understand, the problem i'm having is with the logic. One example ...
5
votes
1answer
174 views

Numbers permutation

Given $n$ numbers and $k$ positions I want the total number of permutations of these n numbers on these $k$ positions if repetition is allowed and if the following two arrangements are considered ...
0
votes
2answers
275 views

There are two integers whose sum and difference are perfect squares

Definition: A positive integer $m$ is said to be a perfect square if there exists an integer $n$ such that $m = n^2$. Write a detailed structured proof to prove that there exist two distinct ...
1
vote
2answers
82 views

Problem of Ages (Problema das Idades)

English: Somebody help me with this challenge? It's very confusing: Today, both me and my younger brother are between $10$ and $20$ years old. Also, our ages are expressed by prime numbers and the ...
2
votes
1answer
104 views

$xy$ itself square in this particular logic

I would like to know the solution or procedure to find the exact analysis/solution of one of my observation. let $x = a^2$ and $y = b^2$, then can we express $xy$ (concatenation of $x$ and $y$) as ...
1
vote
1answer
48 views

Missing one link in logic of basic unique factorization argument

From page 2 of The Prime Facts : from Euclid to AKS by Scott Aaronson : Thus P/A = R/K. But R is less than P, since it’s a remainder from dividing by P. Okay So P/A can’t be in lowest ...
3
votes
4answers
136 views

If $b\mid ca$, then $b\mid a$. Is this true?

My proof: We want to show $b\mid a$ i.e. $a = bn$ for some integer $n$. Since $b\mid ca$, $ca = bm$ for some integer $m$. Substituting for $a$ gives us $c(bn) = bm \Rightarrow b(cn) = bm\dots$ After ...
1
vote
1answer
407 views

Gödel, Escher, Bach: $ b $ is a power of $ 10 $.

I’d like to verify if my formula correctly expresses that a number is a power of $ 10 $, using the $ \sf{TNT} $ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal ...
0
votes
2answers
1k views

Factorial (Proof by Induction)

Prove by induction that $n!<n^n$ for all $n>1$. So far I have (using weak induction): Base Case: Proved that claim holds for $n=2$ Induction hypothesis: For some arbitrary $n>1, n!<n^n$ ...
2
votes
3answers
670 views

Hofstadter's TNT: b is a power of 2 - is my formula doing what it is supposed to?

If you've read Hofstadter's Gödel, Escher, Bach, you must have come across the problem of expressing 'b is a power of 2' in Typographical Number Theory. An alternative way to say this is that every ...
51
votes
2answers
2k views

Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on

Popular mathematics folklore provides some simple tools enabling us compactly to describe some truly enormous numbers. For example, the number $10^{100}$ is commonly known as a googol, and a googol ...
8
votes
2answers
440 views

How can my proof be improved? “Let $n$ be an integer. If $3n$ is odd then so is $n$.”

I am attempting to self-study proof techniques and your criticism of my following proof would be greatly appreciated. Feel free to nitpick minor/trivial things; that's how I'll learn! Edit: I have ...