Search found 76 matches

by Alvaro
Fri Dec 25, 2009 2:07 pm
Forum: Number
Topic: Digit-by-Digit Fibonacci-style Sequence
Replies: 4
Views: 3043

Re: Digit-by-Digit Fibonacci-style Sequence

The length of the cycle formed by Fibonacci mod 10 is 60. Let's say that as c(10)=60. While I was in college I proved, together with a friend of mine, the following facts about c: * If gcd(a,b)=1, then c(a*b) = lcm(c(a),c(b)) * If p is a prime and p=1(mod 5) or p=4(mod 5), then c(p) divides p-1 * If...
by Alvaro
Fri Aug 10, 2007 6:58 pm
Forum: Number
Topic: Grocery Store
Replies: 2
Views: 1184

Re: Grocery Store

About your food for thought: Do you know about the proof that all natural numbers have some special property? Well, assume it isn't so, and there are indeed natural numbers without any special properties. Let n be the smallest natural number without any special properties. Now, that's a very special...
by Alvaro
Tue Jul 17, 2007 6:40 pm
Forum: Number
Topic: "Summarizing" sequence
Replies: 11
Views: 2778

Re: "Summarizing" sequence

For more precise results in the line that rayfill was pursuing: http://en.wikipedia.org/wiki/Look-and-say_sequence Some hackish/fastish C++ code to compute the answer for 60 (not as fast as the reported assembly solution, but in the same order of magnitude): #include <iostream> char a[100000000]="1\...
by Alvaro
Tue May 01, 2007 2:38 am
Forum: Number
Topic: Patterns...
Replies: 7
Views: 2530

I would use "cycle". But I don't know what you are looking for. If you want to find the cycle of one of these functions modulo a small number, it's trivial to make a little program to do so, and if I were to write out these cycles, I wouldn't know what sequences to put there. On this topic, there ar...
by Alvaro
Mon Apr 30, 2007 11:28 pm
Forum: Number
Topic: Patterns...
Replies: 7
Views: 2530

Well, that's a cycle, but it's not a group.
by Alvaro
Mon Apr 30, 2007 10:48 pm
Forum: Number
Topic: Patterns...
Replies: 7
Views: 2530

How does "x^2=" generate a cyclic group?
by Alvaro
Mon Apr 30, 2007 3:55 pm
Forum: Number
Topic: Patterns...
Replies: 7
Views: 2530

Re: Patterns...

What exactly do you mean when you say that "2x=" will create a cyclic group? (or any or the other lines, for that matter)
by Alvaro
Sat Mar 10, 2007 2:12 am
Forum: Number
Topic: Three squares make a sixth power
Replies: 8
Views: 3239

It's still not interesting... If you set

a=n3
b=anything
c=b

then you get

n6 = a2 + b2 - c2

Maybe c cannot be equal to a or b?
by Alvaro
Wed Feb 21, 2007 3:55 pm
Forum: Number
Topic: Greatest Common Factors
Replies: 5
Views: 2332

The usual axioms for probabilities on infinite sets are Kolmogorov's axioms: http://en.wikipedia.org/wiki/Probability_axioms The notion of "probability" over the naturals that you seem to be using is the following. Given a subset A of the natural numbers, one can count what fraction of the first n n...
by Alvaro
Wed Feb 21, 2007 2:38 pm
Forum: Number
Topic: Greatest Common Factors
Replies: 5
Views: 2332

You can do the same thing for 3 or 5 integers, and although there is no closed-form solution in those cases, it shouldn't be hard to compute a few decimal places.

Also, note that "probability" might not be the right word to use here, since we are not satisfying all the axioms.
by Alvaro
Wed Feb 21, 2007 3:37 am
Forum: Number
Topic: Greatest Common Factors
Replies: 5
Views: 2332

90/pi^4, which is around .92394
by Alvaro
Thu Feb 01, 2007 2:40 am
Forum: Number
Topic: Group Theory
Replies: 5
Views: 2078

I haven't done this kind of thing in more than 10 years, but I think you define inverse as something that is the inverse on both sides. As for the neutral element, you can prove that if the group has a neutral element on one side, it is also an element on the other side.
by Alvaro
Mon Jan 29, 2007 2:02 am
Forum: Number Theory
Topic: A Planet Problem
Replies: 7
Views: 2946

... If no other planet looks at either A or B, you can remove them from the picture and use induction. There is one more possibility to consider (and eliminate) : ... Q [map] A [map] B [map] C ... No. Maybe I didn't express it clearly enough. A and B are the planets such that the distance between t...
by Alvaro
Sun Jan 28, 2007 7:35 am
Forum: Number Theory
Topic: A Planet Problem
Replies: 7
Views: 2946

If you consider the two planets A and B that are closest to each other, A looks at B and B looks at A. If some other planet looks at either A or B, one planet is being watched more than once, so there must be an unwatched planet. If no other planet looks at either A or B, you can remove them from th...
by Alvaro
Tue Jan 23, 2007 3:37 am
Forum: Number
Topic: A prime property
Replies: 3
Views: 2072

We need to find two quadratic residues modulo p that add up to -1. There are exactly (p-1)/2 quadratic residues modulo p, excluding 0. If -1 is a quadratic residue, we are done. If (p-1)/2 is a quadratic residue, we are also done. Now we consider the following (p-3)/2 sets: {1,p-2}, {2,p-3}, ..., {(...
by Alvaro
Fri Jan 19, 2007 7:01 pm
Forum: Game Theory
Topic: Grid Logic
Replies: 4
Views: 3855

I would try something relatively simple, like breadth-first search.
by Alvaro
Fri Dec 29, 2006 6:18 pm
Forum: Number Theory
Topic: smallest multiple of 17 (urgent! )
Replies: 6
Views: 3544

perl -e 'do{print 1;$mod=($mod*10+1)%17;}while($mod);print "\n";'
by Alvaro
Thu Nov 23, 2006 3:09 pm
Forum: Number
Topic: A Binary Number Puzzle
Replies: 3
Views: 2498

I think I have a simple solution. Consider Any five q-digit binary numbers such that there is no place with a common digit for all the numbers. We are going to compare all pairs of numbers, and there are 10 pairs. That means that we are going to compare 10*q pairs of bits. How many matches will we g...
by Alvaro
Fri Nov 17, 2006 1:44 pm
Forum: Number
Topic: A Quadratic Divisor Puzzle
Replies: 3
Views: 2297

Steps for a proof:

* n must be odd, because perfect squares have an odd number of divisors.
* d(a*b)=d(a)*d(b) if gcd(a,b)=1.
* If p is an odd prime, p2k+1 [ge] d(p2k+1)2 = (2k+1)2, and the equality only happens when p=3 and k=1.
by Alvaro
Wed Nov 15, 2006 2:51 am
Forum: Number
Topic: It just doesn't add up?!
Replies: 1
Views: 1554

Re: It just doesn't add up?!

neonash7777 wrote:So we have (3*$9)+$2 which is suppose to equal $30
No, it's (3*$9)-$2 which should equal $25, which is the price of the rooms,