Resources tagged with: Number theory

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There are 22 results

Broad Topics > Numbers and the Number System > Number theory

An Introduction to Number Theory

Age 16 to 18

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions)

There's a Limit

Age 14 to 18 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Pythagorean Golden Means

Age 16 to 18 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Modulus Arithmetic and a Solution to Differences

Age 16 to 18

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

A Little Light Thinking

Age 14 to 16 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Sums of Squares and Sums of Cubes

Age 16 to 18

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

An Introduction to Modular Arithmetic

Age 14 to 16

An introduction to the notation and uses of modular arithmetic

Really Mr. Bond

Age 14 to 16 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

More Sums of Squares

Age 16 to 18

Tom writes about expressing numbers as the sums of three squares.

Euler's Squares

Age 14 to 16 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Data Chunks

Age 14 to 16 Challenge Level:

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . .

Number Rules - OK

Age 14 to 16 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

Mod 7

Age 16 to 18 Challenge Level:

Find the remainder when 3^{2001} is divided by 7.

Always Perfect

Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

The Public Key

Age 16 to 18 Challenge Level:

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

Ordered Sums

Age 14 to 16 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . .

2^n -n Numbers

Age 16 to 18

Yatir from Israel wrote this article on numbers that can be written as $2^n-n$ where n is a positive integer.

Never Prime

Age 14 to 16 Challenge Level:

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Novemberish

Age 14 to 16 Challenge Level:

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Diophantine N-tuples

Age 14 to 16 Challenge Level:

Can you explain why a sequence of operations always gives you perfect squares?

Binomial Coefficients

Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.