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'?
- Square Root of a Prime [07/14/1999]
Suppose p is a prime number. Show that sqrt(p) is irrational.
- The Square Root of i [05/25/1997]
What is the square root of i?
- The Square Root of n! [10/14/1998]
For what natural numbers n is the square root of n! an integer?
- Square Roots in Binary [10/03/2000]
Can you show an example of taking the square root of a binary number?
- Squares in an Infinite Factorial Series [11/23/2001]
How many perfect squares appear among the following numbers: 1!, 1!+
- Square Triangular Numbers [11/22/2002]
Is there an equation for square triangular numbers?
- Stirling Numbers [05/26/1999]
Can you show how to evaluate Stirling Numbers of the first and second
- Stirling Numbers of the Second Kind, Bernoulli Numbers [05/29/2001]
Sk = 1^k+2^K+3^k+...+n^k. Find Sk as a formula.
- Stirling's Approximation [05/16/2001]
Is there a way to get the answer to a factorial without having to
multiply out all the numbers?
- Stones, Prime Powers, Induction Proof [01/23/2001]
A heap of 201 stones is divided in several steps into heaps of three
- Subsets and Greatest Common Divisor [03/26/1999]
A question on subsets and another on greatest common divisor (GCD).
- Subsets of Real Numbers and Infinity [08/22/2001]
Am I correct in saying that both the whole number set and the integer set
have an infinite number of numbers within them, and therefore are of the
- Subtracting Two Numbers of Like Base [10/21/2004]
I'm having trouble with the idea of how to borrow when I am
subtracting two numbers in a base other than base ten. Can you help?
- Subtraction Puzzle [08/18/2002]
For numbers A, B, C, and D, subtract A from B, (or vice-versa; you
must be left with a whole number, not a negative one). Repeat with
B and C, C and D, and D and A. After about 6 steps, you will always
end up with 0000. The puzzle is to get as many steps as possible.
- Subtraction Using Nine's and Ten's Complements [05/27/2000]
How does subtraction using the "method of complements" work? Why does it
give the correct answer all of the time?
- Summing a Binary Function Sequence [07/16/1998]
How do you compute the sum of B(n)/(n(n+1)) from 1 to infinity, where
B(n) denotes the sum of the binary digits of n?
- Summing Activity Leads to a Mean of e [04/01/2005]
I asked my students to keep adding random integers from 1 to 100 until
the sum exceeded 100. We then found the average number of terms
added. The answer seems to be e. Why? The more we do it, the
closer we get.
- Summing a Series Like n*(n!) [10/28/2001]
How can I add up a series like 1*1! + 2*2! + 3*3! ... n*n! ?
- Summing Consecutive Integers [08/30/1998]
Express 1994 as a sum of consecutive positive integers, and show that
this is the only way to do it.
- Summing Four Variables, Given Something of Their Factors [02/20/2013]
What can you determine about the four variables in an equation, given information
about the factors of three of them? By decomposing positive integers with even
numbers of factors into products of primes, Doctor Greenie starts to unpack this
puzzle, case by case.
- Summing n^k [11/24/1998]
Is there a general formula for summing the n^k, where k is a positive
- Sum of 1/Sqrt(i) [11/20/2000]
What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show
me the proof by induction?
- Sum of a Pair — and of the Pair's GCF and LCM [12/09/2012]
A student struggles to identify a pair of positive integers, given their sum as well as the
sum of their greatest common factor and least common multiple. Doctor Greenie
applies some algebra and factorization to turn the problem into a Diophantine
- Sum of Consecutive Odd Integers [07/27/2001]
Given an integer N, can N can be written as a sum of consecutive odd
integers? If so, how can I identify *all* the sets of consecutive odd
integers that add up to N?
- Sum of Digits Divisible by 11 [08/16/1999]
Can you prove that in a sequence of 39 consecutive natural numbers there
exists at least one number such that the sum of its digits is divisible
- Sum of Digits of Multiples of Nine [08/12/2004]
Can you prove that if you add the digits of any multiple of nine, then
add the digits of that result, and keep going, you eventually wind up
with 9? For example, 99 => 9 + 9 = 18 => 1 + 8 = 9. Why does it work?
- Sum of Distinct Fibonacci Numbers [05/06/2001]
How do you show that every positive integer is a sum of distinct terms of
the Fibonacci sequence?
- Sum of Divisors Proven [08/10/2013]
A student stumbles over the formulas for divisor numbers and sums. Doctor Peterson
outlines the proof of the latter, with examples.
- Sum of First n Cubes, First n Squares [11/18/2002]
Is there a shortcut to find (1^3-1^2)+(2^3-2^2)+(3^3-3^2)... (15^3-15^
- Sum of First n Natural Numbers [12/03/2005]
Factorial refers to the product of the first n natural numbers. Is
there a name and symbol for the SUM of the first n natural numbers?
- Sum of Integers [07/03/2001]
How many integers are 13 times the sum of their digits?
- Sum of Numbers from 1 to n [01/09/2003]
Is there a formula for calculating the summation of numbers from 1
- Sum of Powers of 2 [08/28/2001]
I want to derive a formula for the sum of powers of 2.
- Sum of Squares of Two Odd Integers [10/26/1999]
How can I prove that the sum of the squares of two odd integers cannot be
a perfect square?
- Sum of Twin Primes [09/04/2003]
Given that a and b are two consecutive odd prime integers, prove that
their sum has three or more prime divisors (not necessarily distinct).
- Sum of Two Cubes [01/12/2002]
Find the smallest number that can be expressed as the sum of two cube
numbers in two different ways.
- Sum of Two Different Primes [02/22/2002]
Can the sum of two different primes ever be a factor of the product of
- Sum of Two Squares [12/04/1997]
What is the smallest number that can be expressed in twelve different
ways as the sum of two squares?
- Sum of Two Squares [05/26/2003]
Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356,
337, 329, 311, 300]?
- Sum of Unit Fractions [07/17/2001]
By induction, prove that every proper fraction p/q with p less than q can
be written as a finite sum of distinct reciprocals of positive integers.