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High School Archive

Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

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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.

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 is rational?

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.)

Last Digits and High Exponents [09/26/1998]

Calculate the following expressions without a calculator: 3^1000 and 7^ 134. In each, what is the final digit?

Last Four Digits of 5^64 [03/27/2001]

How can I find the last four digits of 3^125 or 5^64?

Last Four Digits of the Fibonacci Numbers [05/06/2001]

Show that there is a number ending with four zeros in the Fibonacci sequence; prove that the Fibonacci sequence has a cycle for the last four digits with a length of 15,000.

Least Common Multiple [10/26/1999]

What is the smallest number that is divisible by the numbers 1 through 10?

The Limit of (1+1/x)^x As x Approaches Infinity [02/17/1998]

How Euler calculated e, and what it has to do with the equation (1+1/x)^x.

Line and Unit Circle; Pythagorean Triples [04/16/2001]

If (X,Y) is a point in the 1st quadrant on the unit circle and m is the slope of the line passing through (X,Y) and the point (0,-1), how can I express the coordinates (X,Y) in terms of m? Can this be used to generate Pythagorean triples?

Linear Diophantine Equations [11/27/1997]

... how was the t-variable introduced, and what is the general method?

Linear Proof [11/07/2001]

We say that f is linear provided that for every x, y in its domain, f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the set of real numbers), then f is defined by f(x) = cx for some c belong to R.

Linear Recurrance Relations [08/10/2004]

Is there a general approach to taking a pattern that is defined recursively and finding an explicit definition for it?

Logarithms and Base E [11/20/1997]

Why is the base of a natural logarithm "e" - how did "e" receive a value of 2.17... ?

Long Division in Base 36 [11/05/2003]

I am trying to use an 'imaginary' system that is base 36 (ie. 1, 2, 3, 4, 5, 6, 7, 8, 9, A=10, B=11, ..., Z=35) and I am struggling to divide one number in this system by another using long division.

Long Division in Binary [05/16/2000]

How can you divide 1011 base 2 by 11 base 2?

Long Division in Binary [06/06/2000]

What is the algorithm for binary division? Can you show me some examples?

Lowest Integer That Can't Be Made [04/05/2004]

You have an unlimited number of 'a' cent stamps and 'b' cent stamps, where both a and b are relatively prime positive integers. I am interested in finding a postage value above which any postage amount can be made. I need to find a formula involving a and b, that will give the cutoff point for any such pair of numbers a and b.

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