2
votes
1answer
121 views

Are there any “obviously” true propositions in number theory?

After all efforts spent on wrong proofs of famous number theory conjectures and theorems like Goldbach's or Fermat's last theorem, could one find some simple statements (might be correct ones) whose ...
1
vote
2answers
63 views

Query about Reductio Ad Absurdum

If we use the method of contradiction(i.e.Reductio Ad Absurdum), and if one of our assumptions is wrong, does that mean that all our assumptions are wrong and is the statement or hypothesis proved?
4
votes
3answers
113 views

Diophantine equations and Hilbert's 10th Problem, how did MRDP do it?

I'm having a bit of trouble understanding the Wiki explanation of MRDP's (Matiyasevich, Robinson, Davis, Putnam)'s Theorem, which explains that Hilbert's 10th problem is unsolvable. The MRDP ...
0
votes
1answer
32 views

Proving that operations give equal results given equal inputs

I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as ...
1
vote
3answers
108 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
45 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
184 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
308 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
86 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
106 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
443 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 ...
1
vote
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
822 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 ...
52
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
452 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 ...