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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'?
 Cubes as Differences of Squares [07/04/2002]

Prove that the cube of any positive integer is equal to the
difference of the squares of two integers.
 Cute Numbers [08/29/2003]

I have found two definitions of a cute number. Which is correct?
 Cyclic Redundancy Check [06/26/2002]

I understand how cyclic redundancy checks work, but I fail to see how
appending zeros to the message string (before the division) provides an
advantage.
 Dates that Read the Same Backwards and Forwards [02/02/2010]

A student sees a palindrome in the date 01 02 2010, and wonders how to
generate all such palindromic dates. Building on another math doctor's
work with date arithmetic, Doctor Carter shares a program written in C,
then goes on to explain the purpose of each line of code.
 Decimal Expansion of a Reciprocal [10/23/2001]

1/(X + Y + Z) = 0.XYZ.
 Dedekind Cuts [10/23/1996]

What is a Dedekind cut?
 Defining 0/0 [01/29/2001]

I convinced my teacher that 0/0 must be defined, since our math laws say
that anything divided by itself equals 1. Shouldn't 0/0 = 1?
 Definition of Floating Point Data [07/02/2001]

What are 'floating point data'? How do they differ from an integer? What
are some examples?
 Definitions of Advanced Concepts [11/13/1998]

Can you give me definitions for: Pythagorean Triplets, Principle of
Duality, Euclid's Elements, Cycloid, Fermat's Last Theorem?
 Definitions: Relatively Prime, Proper Factor [9/11/1996]

What does it mean to be relatively prime? What is a proper factor?
 Density Property of Rational Numbers [09/21/2001]

How is the density property of rational numbers proven?
 Deriving Properties of Fractions [08/10/2003]

Derive the rule for multiplying fractions, that a/b x c/d = ac/bd,
using lowerlevel properties of multiplication and rational numbers.
 Determining Factors of a 3998digit Number [08/11/1999]

Let N = 111...1222...2, where there are 1999 digits of 1 followed by 1999
digits of 2. How can I express N as the product of four integers, each of
which is greater than 1?
 Determining If a Large Number is Divisible by 11 [10/22/2003]

I just learned a trick to decide whether a large number is divisible by
11 or not. Why does the trick work?
 Determining Primes by Their Square Roots [06/13/2001]

My problem has to do with determining if a very large number is a prime.
 Diagonal Sum in Pascal's Triangle [04/02/2001]

Find the sum of the reciprocals of the diagonals in Pascal's triangle.
 Difference of Square Numbers [07/18/2008]

Can one number ever be represented as two distinct differences of
squares? Or is every difference of square numbers unique?
 Different Infinities [02/19/1997]

How many different infinities are there?
 Digital Computers and Binary [07/02/2000]

How do digital computers use the binary number system?
 Digit Patterns of the Powers of 5 [09/14/1998]

Why is there a pattern in the last digits of the powers of 5?
 Digits of a Square [05/26/2001]

If the tens digit of a^2 (a is an integer) is 7, what is the units
digit?
 Digits Sums, Mod Proofs, Olympiad Squares, and Equal Roots [12/15/2011]

A student seeks help with four different number theory proofs. Doctors Carter and
Vogler offer observations, textbook recommendations, and other guidance.
 Diophantine by Process of Elimination [01/10/2015]

A teen knows the integer solutions to 2^x  3^y = 5, but seeks proof. Doctor Vogler
steps through an approach that starts by reducing the equation mod m, where neither
of the bases 2 and 3 has an order.
 Diophantine Equations [11/17/1997]

We have searched the Web for information about Diophantine equations.
 Diophantine Equations [06/29/2001]

Find rational x and y such that x^2+x^2*y^2 and y^2+x^2*y^2 are perfect
squares, or, more simply, x^2+x^2*y^2 = m^2 and y^2+x^2*y^2 = n^2, where
n and m are rational numbers.
 Diophantine equations in Number Theory [01/24/2001]

If a and b are relatively prime positive integers, prove that the
Diophantine equation axby = c has infinitely many solutions in the
positive integers.
 Diophantine Equations in Three Variables [10/30/2004]

I need to know how to get positive integer solutions of two
Diophantine equations having three variables. For example: 2x + 3y +
7z = 32 ; 3x + 4y  z = 19. (Give the positive set of triples for the
above equations.)
 Diophantine Equations, Step by Step [10/01/2002]

Find all positive integer solutions to 43x + 7y + 17z = 400.
 Diophantine Equation to Find Perfect Square Values [03/19/2008]

Given a long polynomial such as 4x^4 + x^3 + 2x^2 + x + 1, how can I
find positive integers that would produce a perfect square value when
substituted into the polynomial?
 Direct Conversion of Binary to Octal [05/14/2002]

How can you convert from base 2 to base 8 without going through base
10?
 Discrete Logarithm Problem [10/13/2004]

Given a === b^c mod N. When a, b, and N are given, can we find c?
 Displaying Large Repetends on Small Calculators [05/31/2002]

How can I find a 16digit repetend using an 8digit calculator?
 Distance between Points on a Line [10/02/2002]

When the 10 distances between 5 pairs of points on a line are listed
from smallest to largest, the list reads: 2,4,5,7,8,k,13,15,17,19.
What is the value of k?
 Dividing 29/49 [08/30/1997]

Can I divide 29/49 out until it repeats itself or terminates without
using long division?
 Divisibility by 11: Proof [02/12/2002]

Prove that a positive integer n is divisible by 11 if and only if the
alternating sum of its digits is divisible by 11.
 Divisibility by 37 [11/08/1997]

Take a 3digit number and add to that its "rotation". Prove that the sum
can always be divided by 37.
 Divisibility by 3 in Three Consecutive Numbers [10/07/2002]

With any combination of consecutive natural numbers, why is one
integer divisible by three and why is ONLY one number divisible by 3?
 Divisibility by 8 [04/14/1997]

Show that, if n is a positive integer, then 5^n+2*3^(n1) + 1 is
divisible by 8.
 Divisibility by Three: Proof [07/22/2003]

Why is the sum of the digits of a multiple of 3 divisible by 3?
 Divisibility of Squares of Prime Numbers [02/14/1998]

If p is a prime greater than 3, prove that p^2 leaves a remainder of 1
when divided by 12.
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