The arithmetic tag has no wiki summary.
-1
votes
0answers
16 views
Equating any two decimal numbers using fractional bases? [migrated]
I've always wondered this, but I don't know if it's possible.
Is there a general formula for finding the fractional base that will equate two decimal numbers.
For example:
11 = 5 in base 4
12 = 14 ...
12
votes
1answer
244 views
Geometry of numbers for three by three matrices?
While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...
27
votes
2answers
1k views
A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself
Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
1
vote
1answer
130 views
Infinite board games: sentences about
As a unified approach if we have an ( read any) infinite board game described as $\mathcal{G}$ using a particular axiom set A..
can a sentence be devised in A which automatically answers the basic ...
4
votes
2answers
338 views
Numerical coincidence?
(Nobody's answered this one on stackexchange after several days.)
My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right ...
2
votes
2answers
183 views
Square and reversed integer
For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= ...
32
votes
2answers
1k 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 ...
17
votes
1answer
150 views
Decidability of equality of expressions built using 1,+,-,*,/,^
Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...
10
votes
1answer
355 views
Does any lower bound on proofs of FLT improve Shepherdson 1965?
In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl ...
4
votes
1answer
82 views
What metatheory proves $\mathsf{ACA}_0$ conservative over PA?
Simpson's book shows $\mathsf{ACA}_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}_0$ being conservative over PA is ...
0
votes
0answers
82 views
On the expilicite example of Parshin Construction.
In the Proof of Mordell Conjecture by Gerd Faltings, it is famous that Parshin constructed curve C_P for each Q-rational point P on the given curve C over Q such that genus of C satisfies g(C) > 0.
...
1
vote
1answer
418 views
A formula combining Euler $\phi$ and $\gcd$
Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
0
votes
0answers
78 views
squarefree monomial ideals and simplicial complexes
Let $ I = (f_1, f_2); J = (g_1, g_2), L= (h_1, h_2) $ be monomial ideals in polynomial ring $ S = \mathbb{K}[x_1,...,x_n] $ such that $ f_1, f_2, g_1, g_2, h_1, h_2 \in S, f_1 $ and $ f_2 $, $ g_1 $ ...
2
votes
1answer
175 views
experimental mathematics— how are floating point equations discovered/converted to exact equations
the 2005 AMS article/survey on experimental mathematics[1] by Bailey/Borwein mentions many remarkable successes in the field including new formulas for $\pi$ that were discovered via the PSLQ ...
2
votes
0answers
65 views
Seeking name for an order raising operator in Higher Order Arithmetic.
Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of ...
4
votes
0answers
323 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 $Z$ ...
4
votes
3answers
643 views
Is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
My qeustion is that,
is there any theorem like implicit function theorem in $\mathbb{Q}$ ?
More precisely, let $p(\bar{x},\bar{y})$ be in $\mathbb{Z}[\bar{x},\bar{y}]$ such that in $\mathbb{Q}$, for ...
0
votes
1answer
415 views
A question on Cebotarev's density theorem
Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.
By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes ...
7
votes
1answer
302 views
Unboundedness of primes in bounded arithmetic
Wilkie's well known question asks whether $I\Delta_{0}$ proves the unboundedness of primes. We know that by adding a sentence to $I\Delta_{0}$ which says "the exponential function is total", it is ...
2
votes
0answers
121 views
Is this fragment of arithmetic on $p^{-\infty} {\mathbb Z}$ decidable?
Let $p$ be a prime number. Consider the abelian group $p^{-\infty} {\mathbb Z} = \bigcup p^{-n} {\mathbb Z}$ consisting of rational numbers whose denominator is a power of $p$, under addition.
View ...
3
votes
1answer
144 views
Is there an $E_1$-definition of primality?
Here, $E_1$ denotes the set of arithmetic formulas starting with a bounded existential quantifier, followed by a quantifier-free formula. Is there an $E_1$-formula $\phi$ such that $\phi(n)$ holds
iff ...
8
votes
1answer
303 views
Does higher order arithmetic interpret the axiom of choice?
By second order arithmetic I mean the axiomatic theory $Z_2$, that is Peano arithmetic extended by second order variables with the full comprehension axiom, and not defined semantically using power ...
4
votes
2answers
668 views
Axiom to exclude nonstandard natural numbers
In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the ...
4
votes
0answers
162 views
Modelling the difficulty of mental calculation.
Are you aware of any work that tries to model the difficulty of evaluating a formula mentally (for your average, numerate, person, not a trained mental calculator)?
For instance, evaluating an ...
3
votes
1answer
278 views
Godel 's Ladder: Undecidable PI_N sentences for N =2, 3, …
After Godel's groundbreaking results, a plethora of $\Pi_1^0$ undecidable arithmetical sentences have been found by many authors.
But what about $\Pi_n^0$ for $n=2,3,.....$ ?
There are, to my ...
8
votes
2answers
342 views
Z_2 versus second-order PA
These days, Peano Arithmetic ($PA$) refers to the first-order version of the axioms, where induction is only over formulas referring to natural number variables. Peano's original version of the ...
3
votes
1answer
194 views
Is there exponentiation in “sufficiently large” models of $I\Delta_{0}$?
Let $L_{E}$ be the language of discretely ordered rings together with an extra predicate symbol $E$. The system $A$ consists of the axioms of $I\Delta_{0}$ (basic arithmetic plus induction for bounded ...
1
vote
3answers
432 views
Applicability of Deduction theorem to Primitive recursive arithmetic [closed]
Hello. I already asked the question here. The main point is that I tried to prove in Primitive recursive arithmetic (PRA) the totality of the Ackerman function, and I found, that the single thing ...
3
votes
2answers
380 views
NNO = (first order) PA
Recall the definition of a Natural Numbers Object in a topos, and the first order axioms for Peano Arithmetic. I am more familiar with the first definition than the second, so I cannot tell from the ...
7
votes
4answers
500 views
Reference Request: Non-Standard Models of PA
I am attempting to write an expository paper on non-standard models of PA that is accesible to students taking an introductory graduate course in mathematical logic (covering Godel's incompleteness ...
0
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0answers
246 views
the mathematics of clocks [closed]
I'm searching for an elegant way to increment and decrement around a 'circle' of numbers like a clock.
Assuming the numbers 1,2,3 are my set of numbers on the clockface I want to perform the ...
5
votes
4answers
389 views
parameters in arithmetic induction axiom schemas
The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$. However, since the ...
4
votes
1answer
231 views
Cardinality of the set of elements of fixed order.
Let us consider the group $G:=\mathbb{Z}_N^a$ (the product of the cyclic group with $N$ elements with itself $a$ times). Suppose we are given a number $m$ that divides $N$.
I would like to know how ...
2
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0answers
172 views
Algorithm/denominators of elements of a rational affine space
Hi,
I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free ...
1
vote
0answers
385 views
An elementary question in modular arithmetic
Let us fix a positive natural number $N$. When $i$ is a natural number smaller than $N$, coprime with $N$, we let $\mu(i)$ be the unique number in $\{1, \ldots, N-1\}$ that is the multiplicative ...
3
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0answers
237 views
Hochschild-Serre for hypercohomology
I need either a proof or a good reference for the following plausible statement:
Let $S$ be a scheme and let $C$ be a bounded complex of abelian sheaves on $S_{\rm{fppf}}$. Let $S^{\prime}\rightarrow ...
7
votes
1answer
464 views
Nelson natural number objects in a topos (say)
Nelson's predicative arithmetic (survey article) is a very weak system of arithmetic extending Robinson's $Q$ (Wikipedia).
We can have natural number objects in a topos, or even a merely finitely ...
2
votes
2answers
498 views
Natural numbers of great kolmogorov complexity
Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, ...
5
votes
2answers
819 views
Set Theory inside Arithmetics via the Ackermann Yoga
Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of $ZF$-Infinity in $PA$ (see for refs this MO question and here for an excellent ...
0
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0answers
273 views
Why does this work? [closed]
Hi I need to know how the equation below works.
So I have a number line 0 to 21 each cell is divided into 3 units so 0-2 belongs to first unit, 3-5 belongs to the second unit and so on. This 3 unit ...
9
votes
9answers
3k views
Is PA consistent? do we know it?
1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...
10
votes
5answers
1k views
are there infinitely many triples of consecutive square-free integers?
The title says it all ... Obviously, any such triple must be of the form
$(4a+1,4a+2,4a+3)$ where $a$ is an integer. Has this problem
already been studied before ? The result would follow from ...
0
votes
2answers
326 views
Which numbers appear as discriminants of cubics?
I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants.
Clearly, given that they are products of squares, all ...
8
votes
2answers
576 views
The different Branches of Arithmetics
... "and then the
different branches of Arithmetic--
Ambition, Distraction, Uglification,
and Derision."
(Alice in Wonderland, chapter IX: the Mock Turtle's story)
As a child I ...
10
votes
2answers
560 views
Is Robinson Arithmetic biinterpretable with some theory in LST?
Let ZFC$^{\text{fin}}$ be ZFC minus the axiom of infinity plus the negation of the axiom of infinity. It is well-known that ZFC$^{\text{fin}}$ is biinterpretable with Peano Arithmetic. In this sense ...
0
votes
2answers
582 views
Is division considered the mathematical dual of multiplication? [closed]
I'm doing a bit of research for a tech presentation that touches on the subject of mathematical duality. (To be clear, my presentation is not on mathematics or duality, but mentions duality in ...
3
votes
0answers
307 views
example just slightly better than the greedy construction
Roth's theorem provides an estimate for the largest
size of a nonaveraging subset of $\lbrace 1,2, \ldots ,n \rbrace$ (a set of integers is nonaveraging if it does not contain any nontrivial ...
16
votes
2answers
1k views
unboundedness of number of integral points on elliptic curves?
If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...
1
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0answers
521 views
What is the name for(a^2 + b^2 + c^2 +…)/(a + b + c +…)? [closed]
That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.
2
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2answers
196 views
FTA in first order setting
When I took model theory is an undergraduate, early on we wrestled with trying to state the fundamental theorem of arithmetic in the first order language of arithmetic. The problem was that we needed ...
