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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'?

Solving x^y = y^(x - y) for All Natural x, y [11/18/2010]
A student seeks all natural numbers x, y such that x^y = y^(x - y). With chains of reasoning about integer divisibility and exponentiation, Doctor Vogler deduces all three solutions.

Spacing between Prime Numbers [11/08/2005]
Where is the first place that the difference between two consecutive prime numbers exceeds 2000? Is there a formula or general approach to finding such differences without having to just read through lists of known primes?

Splitting a Sum of Integers into Two Equal Sums [09/07/2004]
Given integers {1,2,3,4,...n}, prove that if their sum is even they can be split into two equivalent sums, each equal to half of the original sum, and using each integer once.

The Square and Multiply Method [08/31/1998]
Solve an encryption problem by solving the math function 33815^(81599) (mod 154381).

Square Numbers, All Digits the Same [06/17/2003]
Is there any square number with all the same digits?

Square of an Odd Number [11/12/2002]
True or false: the square of any odd number can be represented in the form 8n+1, where n is a whole number.

Square Root and Sum of Digits [07/25/2008]
I noticed that 81 has the same square root as the sum of its digits since 8 + 1 = 9 and the square root of 81 is also 9. Are there other numbers that have the same property?

Square Root of 2 as a 'Vulgar Fraction' [05/04/2001]
Can the square root of 2 be expressed as a fraction?

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!+ 2!,1!+2!+3!,...1!+2!+3!+...n!?

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 kinds?

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 stones each...

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 same size?

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 integer?

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 equation.

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 by 11?

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^ 2)?

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 through n?

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