Natural numbers Definition and 141 Threads

I was pondering square numbers today, and I noticed something interesting: every natural number contains information to construct a Pythagorean triple. Let me show what I mean for an odd natural number, also using the binomial theorem for square quadratic equations (equations of the form...

fin all natural numbers n that n\sigma (n) \equiv 2(mod\varphi(n))

Hello! Question: if it is asked to prove a statement A(n_1,...,n_k) for all natural numbers n_1,...,n_k, is it actually enough to check its truth by induction on just one of the counters, say n_1?

ok. some rant about definition and semantics. integers are isomorphic ordered pairs of natural numbers (a,b) w/ equivalence relation (a,b)=(c,d) iff a+d=b+c. reals are convergent sequences of rationals, etc. in mathematics, are integers simply isomorphic to the ordered pairs of...

In class we defined convergence as \forall\varepsilon>0\;\;\;\exists\mathrm{N}\epsilon\mathbf{N}\;\;\;\forall\mathrm{n}\geq\mathrm{N}\;\;\;\left|a_{n} \right|<\varepsilon So if a sequence {a_n} of real numbers converge to 0 if for every ε > 0 there is N s.t. |x_n| < ε for n ≥ N... Is the...

Homework Statement Prove that every Natural number is either even or odd. Homework Equations Mathematical Induction Even: n = 2k where k is an integer Odd: n = 2k + 1 where k is an integer The Attempt at a Solution I think I have a relatively complete proof, however, it doesn't...

It might be a silly question but I was just wondering if this was unprovable or false... Thanks guys

I have to prove that for all k,m,n \in N that if m+k = n+k, then m=n. The problem mentions that I must prove this by induction. I did the base case k = 0: If m+0 = n+0, by identity m=n. Then I attempt to show that m+1 = n+1 implies m=n, but I am stuck, I don't see how induction can be...

Homework Statement Let a,b \inR with a < b. and let n \inN where n(b-a) > 1. a) How do you know that such an n must exist? b) Show that there exists m \inZ where a < m/n < b c) Show that there exists some irrational c where a < c < b (Hint:rational + irrational = irrational.) Homework...

If n is a natural number then n2+n+3 is odd. This is what I have and wanted to know if I was doing it right or not: Let n be a member of the natural numbers. If n is even, then n=2k, k member of natural numbers, and n2+n+3 =(2k)2+2k+3 =4k2+2k+3...

Hi, how would you show that 4^(k)+4 * 9^(k) \equiv 0 (mod 5)

Why do some people define the natural numbers as the integers 0,1,2,3... while others define them as the integers 1,2,3... ?

In How many ways can one write a natural number M as a sum of N whole numbers? Consider the two conditions; 1)the numbers appearing in the sum are distinct. 2)the numbers appearing in the sum are not necessary distinct. eg1:eight can be written as a sum of 6 whole numbers as shown below...

1. I was trying to understand the proof of (d/dx) b^x = (ln b)*(b^x) it says: b= e^(ln b) so, b^x= e^((ln b)*x)) So now we use the chain rule: (d/dx) b^x = (d/dx) e^((ln b)*x)) I understand everything so far, but not the next step. It says then that (d/dx) e^((ln b)*x))= (ln...

I don't know if this is more appropriate for the topology forum, but I am learning this in analysis. I am asked to say whether or not Q and N are homeomorphic to each other and to justify why. I am confused as to how to prove precisely that two spaces are homeomorphic, for there are no formal...

Hi, I was wondering if anyone could find a solution to this: d^2-ab=e^2-bc=f^2-ac in the natural numbers where not all variables are equal. I don't know how to make a computer program, but if it takes little time, I would really appreciate if I could have a solution to it.

A nice puzzle I just found (hope it hasn't been posted before): For which natural numbers n does the expression \sqrt {30 + \sqrt n} \ \ + \ \ \sqrt {30 - \sqrt n} yield also a natural number?

Hi, I was reading "Introduction To Set Theory" by Karel Hrbacek and Thomas Jeck and stuck with some logical trap in the proposition that "(\textbf{N},\prec) is a well ordered set" where \textbf{N} is set of all natural numbers. I will try to present the argument briefly to clarify the...

Homework Statement Proof that for n>2 and n is a natural number it holds that \prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<3 and \prod_{k=1}^{n}\frac{k^{2}+2}{k^{2}+1}<\frac{3n}{n+1} Homework Equations The Attempt at a Solution My best approach was to split the product over the...

Homework Statement Prove that the set of all subsets of the natural numbers is uncountable. Homework Equations All of the countability stuff - including Cantor's diagonal argument The Attempt at a Solution I think I have this one figured out, but I was wondering if somebody would...

[SOLVED] Larson 4.1.6 Homework Statement Prove that there are infinitely many natural numbers a with the following property: The number n^4+a is not prime for any number n.Homework Equations The Attempt at a Solution I cannot even think of one such natural number a. :( I need to find some way...

So I'm taking an introductory number theory course as an undergraduate, and this particular "genre" of questions really just has me stumped. Pick a prime p such that p is odd. Now, take various sums up of natural numbers from 1 to p, and show that the results are divisible by p. For...

Homework Statement f(x) = (x+1)/x^2 a)prove that f is bounded in N (N is the set of natural numbers so we have to prove that f(N) is a bounded set) b)find supf(N) and inff(N). c) does f have a maximum or minimum in N? Homework Equations The Attempt at a Solution First I...

I am trying to prove that the set of natural numbers is well-ordered using induction. I am assuming of course that the natural numbers are defined as as the smallest inductive set (of the type specified by the Axiom of Infinity) and that the usual order is defined by n<m iff n\in m. I have an...

Hi, just can't get my head around how to draw these three graphs. Any help appreciated. Thanks In each case below, draw the graph of a function f that satisfies the given property. Give an example of a function f : N -> Z that is bijective/that is injective but not surjective/that is...

Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?

Homework Statement Prove that: 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all n where n is a natural number Homework Equations Proof by induction: a) p(1) is true b) assume p(k) is true then prove p(k+1) for it The Attempt at a Solution I gave this a try for a while but...

Natural Numbers - Pi, Log, ... Greetings, I'm far from an expert on math and I wanted to appeal to smarter minds to help compile a list of numbers with non-repeating chaotic decimals: Here's the first 2 (the only 2 I can think of) Pi e I'd also include the order of prime numbers...

Is there a way to find the,"Sum of the square roots of the first n natural numbers"?

How do I show that \sum_1^n\frac{1}{k} is not an integer for n>1? I tried bounding them between two integrals but that doesn't cut it. I know that \sum_1^n\frac{1}{k}=\frac{(n-1)!+n(n-2)!+n(n-1)(n-3)!+...+n!}{n!} but I can't get a contradiction.

i have a few question, that i hope they will answered. 1) let w={0,1...,n,..}={0}UN, and let f:wxw->w such that the next requirements apply: a) f(0,n)=n+1 b) f(m+1,0)=f(m,1) c) f(m+1,n+1)=f(m,f(m+1,n). i need to prove that for every n,m in w, the next statement applies: f(m,n)<f(m,n+1)...

Any nice proofs for this? 2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1} I hope the tex came out alright. have fun! ps. n is any natural number.

Could Someone help with this question ? What is the probability that a number of the set \Omega, first 120 natural numbers {1,2,3, ... , 120}, picked at random is not divisible by any of the number 3, 4, 6 but is divisible by 2 or 5 ? Thanks

Could Someone help with this question of probability ? What is the probability that a number of the set \Omega, first 120 natural numbers {1,2,3, ... , 120}, picked at random is not divisible by any of the number 3, 4, 6 but is divisible by 2 or 5 ?

Hello. I am working on this problem 0=0.002*e^-(0.005/2R) I am supposed to find to find "R". The only way I know how to do this gives me 0... but I know that it's not the answer. Got any tips?

Is it possible to express all natural numbers greater than 2 as the sum of N unique prime numbers? For example, 6 = 2 + 3 and 18 = 13 + 5.

How can one write any natural number n in one and only one way as 1 less than a power of 2 times an odd number? n=2^f(n)(2g(n)-1)-1 where the red minus is the modified difference function, i.e. x-y =x-y if x>=y or x-y=0 if x<y

In semi-response to Organic's post I thought I'd half take up one of his challenges: Let S = NxT be the product of the natural numbers, N, with the set of all rooted finite trees (or directed graphs satisfying the obvious conditions), embedded in the plane, with the natural ordering on the...

I would greatly appreciate if someone just at least put me in the right direction with this. I have to prove this: Prove that of a,b,c are natural numbers, gcd(a,c) = 1 and b divides c, then gcd(a,b) = 1.

If set A consisted of all even natural numbers (i.e. 2, 4, 6...) and set B consisted of all odd natural numbers (i.e. 1, 3, 5...), then what is the result set of: A intersection B Would it just be the empty set since no two natural numbers can be even and odd?

We know that the sum of the first n natural numbers is n(n+1)/2 Can we express the product of the first n natural numbers without using the factorial symbol? It is possible to write a factorial as a sum. Any idea what it would look like?