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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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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 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 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 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?

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 those primes?

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.

Sums Divisible by 11 [10/10/2001]

Why is the sum of a number with an even number of digits and that same number written in reverse always divisible by 11?

Sums of Consecutive Integers [01/03/2001]

How many different ways can 2000 be expressed as the sum of two or more consecutive positive integers?

Sums of Consecutive Integers [02/04/2001]

What numbers can be expressed as the sum of a string of consecutive positive integers?

Sums of Consecutive Integers with Digital Sums [01/27/2004]

Find all sets of positive consecutive integers that sum to 100, and whose digits sum to greater than 30.

Sums of Consecutive Numbers [06/20/2002]

In what way(s) can 1000 be expressed as the sum of consecutive numbers?

Sums of Consecutive Odd vs. Even Integers [04/15/2002]

Can the sum of two consecutive even integers ever equal the sum of two consecutive odd integers? Why or why not?

Sums of Consecutive Positive Integers [03/02/2001]

Why are the powers of 2 the only numbers you cannot get as the sum of a series of consecutive positive integers?

Sums of Sets of Prime Numbers [01/07/2003]

Given several sets of prime numbers, use each of the nine non-zero digits exactly once. What is the smallest possible sum such a set could have?

Sums of Square Integers Puzzle [07/01/2002]

How many numbers from 1-100 can be expressed as the sum of the squares of two positive integers?

Sums of Three Squares [05/18/1998]

What numbers cannot be expressed as the sum of three squares?

Synthetic Division [11/13/1997]

Why does synthetic division work?

System-Level Programming and Base 2 [05/03/2001]

In computer programming, I have a result that contains several values, always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304, how can I tell if 2^3 exists in 304?

Systems with More Variables than Equations [12/11/2002]

How do I solve a system that has three variables and only two equations, such as 187y + 98x + 45z = 48 and 2y + 9x + 3z = 198?

Taylor Expansion [11/21/2001]

Can you give me the proof of this statement: arcsin(x) = x + 1/2 (x^3/ 3) + (1/2)(3/4)(x^5/5) + (1/2)(3/4)(5/6)(x^7/7) + ... The basis of the calculation is a Taylor series.

A Theorem to Find Lattice Points [6/1/1996]

What are the conditions under which the line ax+by=c will contain lattice points?

Three Number Theory Questions [10/25/1999]

Find the sum of the digits in 4444^4444; find how many times the digit 1 occurs from 1 up to 10,000,000,000; find 3 integers greater than 5^100 that are factors of (5^1985)-1.

TI-86 Base Conversion Program [03/19/2002]

I have finished writing a program that can convert any number in any base (one-ten) to base ten. Now I am writing a program to convert any number in base ten to a given base.

Towers of Hanoi [10/08/2000]

Can you prove the formula 2^n - 1 for the least number of moves it takes to move all n discs to another peg in Towers of Hanoi?

Trailing Zeros and Zero Factorial [04/07/1998]

How many trailing zeros are there for 100! ?

Transfinite Arithmetic [10/28/1997]

What is transfinite arithmetic? I pretty much know what it means, but I am having trouble applying it to aleph-null.

Transfinite Numbers [11/07/1997]

I know that Georg Cantor discovered transfinite numbers, but what are they?

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