Integers Definition and 474 Threads

Could somebody explain with due brevity why/how the set of p-adic integers is homeomorphic to the Cantor set less one point for any prime p? This is a quote from Wikipedia:Cantor Set: "The Cantor set is also homeomorphic to the p-adic integers, and, if one point is removed from it, to the...

I was working with Fourier series and I found the following recursive formula for the zeta function: \frac{p \\ \pi^{2p}}{2p+1} + \sum_{k=1}^{p} \frac{(2p)! (-1)^k \pi^{2(p-k)}}{(2(p-k)+1)!} \zeta(2k) = 0 where [itex]\zeta(k)[/tex] is the Riemann zeta function and p is a positive integer. I...

While responding to another thread about summing nth powers of integers, I came up with what might be a new method. There is an old one (Faulhabers formula) http://mathworld.wolfram.com/FaulhabersFormula.html" , but mine seems to be considerably simpler. Most likely, someone has already...

Find the quotient field of a ring of Gaussian integers?

Prove that summation of n(n+1)/2 is true for all integers. Why is my proof not valid? Could someone explain to me how this is not a valid proof of the summation of "i" from i=1 to n: n(n+1)/2 Show for base cases: n=1: 1(1+1)/2=1 n=2: 2(2+1)/2=3 n=3: 3(3+1)/2=6 ... inductive...

I have a found a hypothesis which I would like you to look at, and perhabs (dis)prove.. ----------------- All integers (n) bigger than 2 (3, 4, 5, 6, ...) be descriped as: n = (p_1 * p_2 * ...) + k where all p and k are primes, but also include 1. Notice that k < (p_1 * p_2 * ...), and...

a = 238000 = 2^4 x 5^3 x 7 x 17 and b = 299880 = 2^3 x 3^2 x 5 x 7^2 x 17 is there an integer n so that a divides b^n if so what is the smallest possibility for n

Hi everybody, We define multiplication as an operation with these properties : a(b+c)=ab+ac and (a+b)c=ac+bc ,a*0=0 and a*1=a with a,b,c natural numbers and of course the two properties Zurtex mentioned ab=ba and a(bc)=(ab)c-I "forgot" to mention them because I didn't use them in what is...

Hey! Can someone please give me a hint on this :rolleyes: Prove: \sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} What I've got so far: Let P(n) be the statement: \sum_{n=1}^n i^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} Let n=1 we get; \sum_{n=1}^1 i^4 =...

Primes in ring of Gauss integers - help! I'm having a very difficult time solving this question, please help! So I'm dealing with the ring R=\field{Z}[\zeta] where \zeta=\frac{1}{2}(-1+\sqrt{-3}) is a cube root of 1. Then the question is: Show the polynomial x^2+x+1 has a root in F_p if...

We have a question that asks to find the number of integers between 1 and 100 that are divisible by 2,3 or 5. So i use the sum rule let E,F and G be the the integers. so, n(EUFUG) = n(E) + n(F) + n(G) - n(EnF) - n(EnG) - n(FnG) but this doesn't work. it gives me the answer of 70 but the...

Let a, b, c and d be 4 distinct integers. Find the smallest possible value for 4(a^2 + b^2 + c^2 + d^2) - (a+b+c+d)^2 and prove that your answer is correct. I got 20 as the smallest answer. Thats when u have a, b, c, and d as 4 consective integers, but i can't prove my answer. Can anyone...

Find the 2002nd positive integer that is not the difference of two square integers. I have idea for the answers, but there are two.

How many positive integers less than 500 have exactly 15 positive integer factors? I know the answer, but not sure it. Can you give me the answer ?

I have to either give an example or show that no such function exists: A real valued function f(x) continuous at all irrationals and at all the integers, but discontinuous everywhere else. I think such function exists and I would define it as follows: f(x) = 0 if x is an irrational...

Ok, I haven't done maths for a few years now and I've been set the following question: The sum of the first integers is given by: Sum(n) = 1+2+3+4 ... +n = n(n+1)/2 Find similar formulae for Even(n) = 2+4+6+8 ... +2n Odd(n) = 1+3+5+7 ... +(2n-1) Now the formulae I have come...

Find all positive integers c such that it is possible to write c = a/b + b/a with positive integers a and b. Please help me :smile:

Does anyone have an idea why this expression in Matlab with integers does not give zero as an answer: >> 1019 / 250 * 250 - 1019 ans = -1.1369e-013

is there a way to prove that n consecutive integers is always divisible by n!? thanks in advance

hi i am new to THIS place here but i do put posts on the number theory site as well. i am in need of direction and have no idea where to turn. i need help w/ two ?'s and they are... how many pos int. <1000 are NOT divisible by 12 or 15? prove the if the sum of two consec. int. is a...

Problem: Write the number 3.1415999999999... as a ratio of two integers. In my book, they have a similar example, but using 2.3171717... And this is how they solved that problem. 2.3171717... = 2.3 + (17/10^3) + (17/10^5) + (17/10^7) + ... After the first term we have a geometric series...

I understand how to solve: a=12mod7 => a = 5, I think, however, how do you solve for a=7mod12 ? Stumped :eek:

The arithmetic mean of a set of nine different positive integers is 123456789. Each number in the set contains a different number of digits with the greatest value being a nine-digit number. Find the value of each of the nine numbers .

Induction Hypothesis: In fact pa is true for all integers n greater than a particular base value and you should complete the proof given below to use the principle of mathematical induction to prove this. pa : n-2 < (n^2 – 3n)/12 Base case is n = 14 Because: n-2 < (n^2 – 3n)/12...