All Questions
Tagged with theories-of-arithmetic nt.number-theory
20 questions
16
votes
2
answers
713
views
Is (Z,+,0,1,P2,P3) decidable?
Is Presburger arithmetic, augmented with two unary predicates P2, P3, for powers of 2 and powers of 3 respectively, decidable?
I know that adding just one of P2, P3 to Presburger keeps it decidable, ...
- 16.6k
14
votes
3
answers
2k
views
Which recursively-defined predicates can be expressed in Presburger Arithmetic?
In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for ...
- 4,713
17
votes
3
answers
3k
views
Gödel's Incompleteness Theorem and the complexity of arithmetic
In How complicated can structures be? Jouko Väänänen says:
The guiding result of mathematical logic is the Incompleteness Theorem of Gödel,
which says that the logical structure of number theory ...
- 1,446
2
votes
0
answers
79
views
Which sets of natural numbers are "lambda-analytic"?
Begin with a bit of notation. Let $t = t_0, \ldots, t_d$ be a finite sequence of real numbers. Define
$$\lambda^t(x) = x^{t_0} \log(x)^{t_1} \log(\log(x))^{t_2} \cdots.$$
for all real numbers $x \in ...
- 13.3k
8
votes
1
answer
544
views
How arithmetical is algebraic exponentiation?
Suppose $K$ is an exponential real closed field, i.e. there is an isomorphism, say exp, between the additive group of $K$ and the multiplicative group of its positive elements.
Assume further that $Z$ ...
- 47.1k
11
votes
2
answers
967
views
How much induction does a p-adic valuation need?
Recently I learned a nice constructive proof of the irrationality of $\sqrt{2}$, which uses the 2-adic valuation of an integer: the count of how many times a number is divisible by 2. The valuation ...
- 44.4k
17
votes
1
answer
2k
views
What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
- 52.3k
14
votes
0
answers
654
views
Reverse Mathematics of Euclid's theorem
Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
CommunityBot
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2
votes
2
answers
1k
views
Are there non-commutative models of arithmetic which have a prime number structure?
Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...
- 12.4k
57
votes
2
answers
7k
views
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
CommunityBot
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3
votes
1
answer
403
views
Simplest PA theorem whose proof requires encoding of sequences even though the statement itself doesn't
What is the simplest number-theoretic theorem whose proof requires exponentiation or finite sequences/sets (so any proof in Peano Arithmetic would need to use encodings of such things using e.g. Gödel'...
CommunityBot
- 1
9
votes
2
answers
1k
views
divisible by all standard prime numbers
This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...
CommunityBot
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9
votes
1
answer
926
views
Application of the Riemann hypothesis and the ABC conjecture to independence results
In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...
2
votes
1
answer
347
views
Elementary proof of bounds on factor polynomials
The question Getting a bound on the coefficients of the factor polynomial got very nice answers on Gelfond's theorem. But for work on proof theory of arithmetic I want a proof in arithmetic. The ...
CommunityBot
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3
votes
2
answers
328
views
Efficient representations of natural numbers via arithmetical expressions
A given natural number $n \in \mathbb{N}$ has many representations
as expressions mixing other natural numbers and the operators and punctuation symbols
$\{+,-,\times,/,\exp,(,)\}$, where '$\exp$' ...
- 1,545
0
votes
2
answers
2k
views
Gödel, Escher, Bach: b is a power of 10. [closed]
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 ...
- 53.3k
0
votes
0
answers
315
views
Definitions for Oddness
In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...
CommunityBot
- 1
5
votes
2
answers
983
views
finite or infinite many quadratic fields embedding into quaternion algebras?
Suppose $H$ is a indefinite quaternion algebra over $\mathbb{Q}$. Are there infinitely many quadratic fields that can be embedded into $H$?
- 65.4k
25
votes
3
answers
3k
views
Composite pairs of the form n!-1 and n!+1
It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$.
Is ...
CommunityBot
- 1
42
votes
7
answers
3k
views
How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
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