An integer (from the Latin integer meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not.
The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold
(
Z
)
{\displaystyle (\mathbb {Z} )}
letter "Z"—standing originally for the German word Zahlen ("numbers").ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
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...
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..
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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 :uhh:
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...
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...
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...
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...