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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'?
 Finding Integer Pairs Whose Product Consists Only of 1's and 0's [10/06/2004]

Given the base10 representation of any integer a, does there exist a
nonzero integer b such that the base10 representation of the product
ab contains only ones and zeros?
 Finding Integer Solutions to a^b = b^a [02/02/2005]

Find all positive integers a and b such that a^b = b^a, and prove that
you have found them all.
 Finding Mersenne Primes [02/15/2002]

How do I find the first four Mersenne primes?
 Finding N Consecutive Composite Numbers [02/26/2001]

How can I find N1 consecutive numbers that are not prime for any number
N greater than 1?
 Finding Number of Trailing Zeros in Factorials [01/27/2005]

I noticed a way to compute the number of trailing zeros of a factorial
in base 10 or 6 at your site. But that method doesn't seem to work for
me in bases such as 4, 8, and 16.
 Finding Patterns in Digits [04/29/1998]

How can we find other solutions to problems like 2^5*9^2 = 2592?
 Finding Perfect Cube Factors of 10! [04/12/2005]

To find all perfect cubes that are factors of 10!, you don't need to
actually evaluate 10!.
 Finding Pi [11/14/1997]

What is the quickest algorithm for finding pi?
 Finding Prime Factors of Factorials [11/26/2004]

How many factors of 2 are contained in 100! ?
 Finding Prime Numbers [11/10/1997]

What is the fastest way to determine if a number is prime?
 Finding Products of a Range of Numbers [03/28/2002]

What method or formula is there to solve for the product of a range of
numbers?
 Finding Pythagorean Triplets [02/02/2006]

Without using the standard a = n^2  m^2, b = 2nm, c = n^2 + m^2, how
can you work out Pythagorean triplets? For example, how could you find
triplets where the hypotenuse exceeds one leg by a certain amount?
 Finding Pythagorean Triplets Algebraically [04/13/2002]

Is there a way to find Pythagorean triplets of Type I algebraically?
 Finding Sets of 7 Prime Numbers That Sum to 100 [05/22/2007]

There are 35 different sets of 7 prime numbers that sum to 100. Of
those sets, which has the largest product, and which has the largest
number? I'm using trial and error and it's very frustrating. Is there
a better way?
 Finding Sets of Integers [01/02/2002]

Without computer assistance, find five different sets of three positive
integers such that and k is less than and m is less than n, and 1/k + 1/n
+ 1/m = 19/84.
 Finding the Exponent with a Modulus [05/24/2003]

I am trying to work out k in the following question: 23^k = 201545
(mod 900001).
 Finding the Last Digits of a Large Exponential [01/01/2005]

What are the last five digits of 7777777^7777777?
 Finding the Sum of the Factors of a Number [07/25/2007]

Is there a formula to find the sum of all the factors of a given number?
 Finding the Two Squares [06/11/2003]

One of Fermat's theorems says that every prime number that yields a
remainder of 1 when divided by 4 can be expressed as the sum of two
integer squares (e.g.: 97 = 4^2 + 9^2). This theorem was proven by
Fermat. What methods are known for determining the two squares?
 Find Number Given Divisor and Remainder Information [01/21/2009]

A four digit number N leaves remainder 10 when divided by 21, remainder
11 when divided by 23 and remainder 12 when divided by 25. What is the
sum of the digits of N?
 Find Remainders: 3^2002/26, 5^2002/26 [09/21/2002]

Find the remainders obtained when 3^2002 and 5^2002 are divided by 26;
show that 3^2002 + 5^2002 is divisible by 26.
 Find the Flaw [08/02/2001]

I don't understand where the following proof goes wrong...
 Find the Monster Mod 11, Mod 2310 [05/07/2003]

Find an integer n between 0 and 2309 with the property that 10^10^10^
34 = n (mod 2310).
 Find the Smallest Number... [10/21/1997]

... that has factors of 1, 2, 3, 4, 5, 6, 7, and 8.
 Find the Smallest Number  A Remainder Problem [09/27/2001]

Find the smallest number, M, such that: M/10 leaves a remainder of 9; M/9
leaves a remainder of 8; M/8 leaves 7; M/7 leaves 6; M/6 leaves 5; M/5
leaves 4; M/4 leaves 3; M/3 leaves 2; and M/2 leaves 1.
 Find the Smallest Triangle [05/25/2001]

A triangle has sides whose lengths are consecutive integers. Its area is
a multiple of 20. Find the smallest triangle that satisfies these
conditions.
 Find the Solution: r^2 + s^2 = c. [01/28/2003]

Given c, find a^2 + b^2 = c^2.
 Find the Unknown Base [06/25/2003]

Two numbers are multiplied to equal another number, which generates a
false number statement. What base is it?
 Finite Series and Greatest Integers [03/06/2003]

For n a positive integer, let t(n) denote the number of positive
divisors of n (including n and 1), and let s(n) denote the sum of
these divisors. Prove the following:...
 Finite vs. Infinite [07/10/1997]

If a line segment is a measurable part of a line, why is the number of
points that make up a line segment infinite?
 Finite vs. Rational [9/10/1996]

A right triangle with sides 1 and 2 has a hypotenuse equal to the square
root of 5, which is irrational  it carries on to infinity without
recurring  but the side length of a triangle must be finite!
 First Calculation of E [12/18/1997]

How did Euler first calculate the value of e?
 The 'First to 100' Game [03/12/2001]

Two players take turns choosing any number from 110, keeping a running
sum of all the numbers. The first player to make this sum exactly 100 is
the winner. Is there a surefire way to win this game?
 Fixed Point and Floating Point Numbers [05/19/2000]

What are fixed point or fixed decimal numbers? How do they differ from
floating point numbers?
 Floating 2 [06/12/2001]

Start with the number 2 on the far left side, then float the number 2 to
the far right side; the new number must be three as large as the old
number.
 FloatingPoint Binary Fractions [07/19/1999]

How can you represent fractions such as 12.93 in binary? How do computers
represent such numbers?
 Floor and Ceiling [05/28/2000]

What do 'floor' and 'ceiling' mean in mathematics?
 Forming Palindromic Numbers [12/04/1998]

Can you give me some examples of forming palindromic numbers with
different operations? How many steps would it take?
 Formula for Connection between Rows of Pascal's Triangle [11/15/2003]

Find a formula connecting any (k+1) coefficients in the nth row of the
Pascal Triangle with a single coefficient in the (n+k)th row.
 Formula for Counting Triangles [03/16/2000]

How many equilateral triangles of integerlength sides are in an
equilateral triangle n units on a side?
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