See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Infinity to the Zero Power [04/28/2001]
Does (infinity)^0 equal 1? Why or why not?
- Integer Iteration Function [12/24/2003]
Let X be a positive integer, A be the number of even digits in that
integer, B be the number of odd digits and C be the number of total
digits. We create the new integer ABC and then we apply that process
repeatedly. We will eventually get the number 123! How can we prove
- Integer Logic Puzzle [04/22/2001]
Two integers, m and n, each between 2 and 100 inclusive, have been
chosen. The product is given to mathematician X and the sum to
mathematician Y... find the integers.
- Integer Root Checking [02/18/2003]
Is there a quick way to check whether a number has any roots that are
- Integers and Complex Numbers [02/27/1997]
Do hyper-reals and octonions exist outside complex numbers?
- Integers and Fractions [03/23/2002]
Give an example of a positive integers p,a,b where p/ab and not p/a and
not p/b. Let m, n, and c be integers. Show that if c/m then c/mn.
- Integers into and out of a Polynomial [08/11/2017]
A teen seeks help finding positive integers m, n such that (n^3 + 1)/(mn - 1) yields
only integer values. After some algebraic manipulations, Doctor Vogler breaks the Diophantine puzzle down into seven cases.
- Integer Solutions of ax + by = c [04/03/2001]
Given the equation 5y - 3x = 1, how can I find solution points where x
and y are both integers? Also, how can I show that there will always be
integer points (x,y) in ax + by = c if a, b and c are all integers?
- Integer Solutions to a Cubic Equation [04/11/2005]
Fermat's method of infinite descent is used to show that the cubic
equation (a^3) + (2b^3) + (4c^3) - 4abc = 0, with a, b, and c whole
numbers and without a=b=c=0, has no solution.
- Interesting Diophantine Equation [12/06/2005]
Find all integers x such that x^2 + 3^x is the square of an integer.
- Intersection of Lines [06/28/1998]
n co-planar lines are such that the number of intersection points is a
maximum. How many intersection points are there? ...
- An Introduction to Basic Diophantine Equations [08/27/2007]
A birdcage contains both 2-legged and 1-legged birds, and there are a
total of 11 legs in the cage. Use a Diophantine equation to find all
possible combinations of birds.
- Introduction to Infinity, Limits, and Why 0.999... Equals One [01/12/2008]
I've read the FAQs and other questions sent to you concerning why
0.999... equals one, but I'm still not convinced. I don't entirely
understand infinity and think it creates more problems than it solves.
- Introducution to Algebraic Numbers and Integers [04/15/2008]
While reading about the classifications of complex numbers, such as
real, irrational, and so on, I came upon a reference to 'algebraic
numbers' and 'algebraic integers'. Can you tell me about those?
- Inverse Function for Natural Numbers [6/10/1996]
I've got a question about the function n = 0.5((a+b)^2+3a+b), which is a
one-to-one bijection from pairs (a,b) of natural numbers to single
natural numbers n.
- Inverse Integers Sums, and Some Unit Fractions [08/05/2017]
How many positive integers satisfy 1/x + 1/y = 1/10000? An adult approaches this
problem by starting to solve a simpler version of it: pairs that sum to a smaller
unit fraction. Doctor Greenie introduces Diophantine equations to provide a more
methodical — and more informative — approach.
- Investigation Involving Square Root of 2 [07/09/2001]
How can you explain the fact that (665857/470832)^2 = 2?
- Irrational and Prime Numbers [4/1/1996]
What are irrational numbers; is there a highest prime number?
- Irrationality of e+pi and e*pi [09/24/2001]
I have read that it is unknown whether either E+Pi or E*Pi is an
irrational number. How can we prove that at most one of the two numbers
- Irrationality of Expressions [09/01/2003]
How can you tell whether (3 sqrt(2) - 1) is rational or irrational?
- Irrationality of Pi [01/09/1998]
Is C/d = a rational number if actually measured?
- Irrationality of Root 2 [03/26/1998]
I've heard that there is a way to prove that root 2 is irrational using a
DIRECT method of proof...
- Irrationality Proof [04/26/2001]
I need to show that log 2 base 10 is irrational.
- Irrational Numbers x,y, x^y Rational? [09/28/2001]
Are there any irrational numbers x and y such that x^y is rational?
- Irrational Pi [12/22/2001]
Why must pi be irrational?
- Is -1 Prime? [10/20/2000]
-1 has exactly two factors (1 and -1). Does anyone consider it a prime?
- Is One a Triangular Number? [03/22/2004]
Triangular numbers have that name because if drawn as dots they can form a triangle. But 1 is just a single dot, so it can't be a triangular number, can it?
- Is Zero a Perfect Square? [08/01/2001]
I have been told that zero is not considered a perfect square, yet the
square root of zero is zero...
- Is Zero a Real Number? [09/27/2001]
My friend and I think we have disproved that 0 is a real number.
- Josephus Problem [04/18/2003]
Every other person at a table is eliminated until there is only one
person left. Who is the survivor?
- Knights of the Round Table [07/01/1998]
If x knights are sitting at a round table, and every other one is
removed, who is the last one left sitting at the table?
- Lagrange's Theorem [02/27/2001]
In your archives you show proofs of Lagrange's theorem that every
positive integer can be expressed as the sum of four squares, but is
there an algorithm for identifying which four squares?
- LaGrange's Theorem [02/24/2001]
Please explain LaGrange's Theorem on the number of roots of a polynomial.
- Large-Number Binary Conversion [01/25/2001]
How do you convert very large binary numbers like 2^50 to base 10?
- Large Numbers and Congruences [03/05/2002]
Find the last three digits of the number 11^(11^(11^(11^(11^11))))
written in base seven.
- Large Prime Numbers [12/17/1997]
Is there an algorithm to determine whether a very large number is prime?
- Large Prime Numbers [01/13/2009]
What is the largest prime number less than which all primes are known?
- Largest 7-Digit Number [07/27/2001]
Work out the largest 7-digit number you can applying two rules: every
digit in the number must be able to be divided into the number, and no
digit can be repeated.
- Largest Integer Divisible by All Integers [01/01/2002]
Show that 24 is the largest integer divisible by all integers less than
its square root.
- Largest x, x^2 less than 2 [10/23/1999]
Prove that there is no largest real number x, such that x^2 is less than
2. (Use indirect proof.)