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Welcome to NRICH.

Number

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Whole numbers

• 2 THREE + THREE + FOUR = ELEVEN
• 2 Odd Numbers
• 2 Is 0 even?
• 2 Minus 0
• 3 Is zero even?
• 2 How do I multiply three digit numbers times three digit numbers?
• 2 Long multiplication and division help!
• 2 How many minutes to the millenium?
• 2 Articles: Magic Squares
• 2 Article: Magic squares for special occasions
• 3 Article:A computer program to find magic squares
• 3 Order of operations
• 3 Arithmagons
• 3 Arithmagons
• 3 Alphanumerics (ABCD×4=DCBA)
• 3 Indices - what happens if the index is 0?
• 3 Constructing integers using 4s
• 4 Lewis Carroll's weird number: 142857
• 4 Finite sequences
• 4 Addition of consecutive numbers
• 4 Summing to 100
• 4 1+2+3+...+999+1000
• 4 Maximum product of integers that sum to a fixed number
• 4 Why does -1 × -1 = 1 ?
• 5 Why does 1 + 1 = 2 ?
• 5 Number of Ways of Summing to n
• 5 Number of solutions of x+y+z=n
• 5 Unique representation of Natural Numbers
• 5 Highest Common Factors
• 5 Haggling for Rupees
• 4 Age problem
• 5 Riddle
• 5 Combinations
• 3 The House of Four Fours
• 5 Definition of Induction
• 5 Weak Induction vs Strong Induction
• 5 Proof that there are no uninteresting numbers
• 5 Integer conjecture
• 6 Natural Numbers in the Real World
• 6 How to prove 1+1=2
• 5 Powers of 3 giving any sequence of digits
• 5 Powers without calulators

Number bases

• 3 Article: Learn About Number Bases
• 3 Binary numbers
• 3 Binary Numbers
• 3 Binary
• 3 Adding in binary
• 4 Bases of numbers

Rounding

• 2 Rounding
• 3 Can someone explain significant figures?
• 3 Large numbers to 2 significant figures
• 4 Is 0.8 really the same as 0.80?

Fractions and percentages

• 2 Dividing by 1/2
• 3 Adding fractions
• 3 Percentages
• 3 Calculating percentages with large numbers
• 3 Calculating Percentages: how much bigger?
• 3 Calculating Percentages: increse/decrease
• 3 Fractions: arithmetic and comparing
• 3 Multiplying Fractions
• 3 Recurring fractions
• 3 Converting decimals to fractions
• 4 Articles: Continued Fractions
• 5 ABA/CDC =0.DEFGDEFG...
• 6 Continued fraction for e^x

Large numbers and standard form

• 2 What is 6,000,000,000,000,000,000,000?
• 3 Biggest number in the world
• 3 1000!
• 3 Zeros in 1000001!
• 5 1000001!
• 4 Standard form, and my teacher training maths homework...
• 5 Factorials and Stirling's Formula
• 5 Very Big Number
• 5 Number of digits in 2¹²³⁴

Calculators

• 4 Higher Precedence
• 4 Calculator can't do (-8)^2/3

Measures

• 2 Is midnight a.m. or p.m.?
• 2 Why 360° in a full turn?
• 2 Temperature project
• 4 Height and mass of scale model
• 4 Why is blood alcohol measured in milligrams per millilitre?
• 4 Motivate project: Where in the world am I?
• For speed/distance/time problems, see under Physics

Factors, divisibility and modular arithmetic

• 2 Factors of zero
• 3 Factors of 64152
• 3 Is 0 a very common multiple?
• 3 2¹⁰⁰⁰/3
• 3 Article: Modulus Arithmetic and a solution to 'Differences'
• 3 Article: Divisibility Tests
• 2 Divisibility by 4
• 2 Divisibility by 11
• 3 Pizza Problem
• 3 Divisibility by 6, 4, 7 or 8.
• 3 Divisibility test for 3 and 9: why does it work?
• 3 70×27¹⁰⁰¹+31×38¹⁰¹ divisible by 13?
• 3 School sprinkler problem: is it on or off?
• 3 Great Granddad problem
• 4 Divisibility Rule for 7
• 4 Divisibility by 11
• 4 Divisibility by 7, 11, and 13
• 4 Introduction to modular arithmetic
• 4 Articles: Whole Number Dynamics
• 4 Remainder problems
• 4 Sum of digits divisible by 170
• 4 3x¹⁰-y¹⁰=1991: no integral solutions?
• 4 Gcd of a⁴-b⁴, a,b prime
• 5 Modular arithmetic
• 5 Modular equation systems
• 5 2ⁿ (mod p)
• 5 2¹⁰ⁿ-1 divisible by 11
• 5 X=n-1(mod n) ((n=2,..,10))
• 5 Prove a×b=(a,b)×[a,b] (HCF × LCM)
• 5 Divisibility proofs
• 5 Sum the digits: digital root of 4444⁴⁴⁴⁴
• 5 Units digit of 3, 3³, (3³)³, ((3³)³)³, etc
• 5 Number divisible by product of digits
• 5 Multiple of 9997 with no even digits
• 5 Sum of Digits Divisible by 3
• 5 Proof by induction: 4ⁿ+6n-1 divisible by 9
• 5 Why does 42 divide n⁷-n for all n?
• 5 Are there infinite numbers with four or six factors?
• 5 k⁷/7+k⁵/5+2k³/3-k/105 is an integer
• 5 Chinese Remainder
• 5 Products ending in 1
• 5 Checking accountancy errors
• 6 _nC_p = floor(n/p) (modulo p)
• 6 Fibonacci in clock arithmetic
• 6 Sequence satisfying a_n | n

Square numbers, triangular numbers, etc

• 2 Article: Sums and Products of Digits and S.P. Numbers
• 3 Sum of the first 10 square numbers
• 3 1234*** is a square
• 3 a³ + b³ = c³
• 3 Number chains (happy numbers)
• 3 Articles: Whole Number Dynamics (happy numbers)
• 3 Triangular Numbers
• 4 Pentagon numbers and triangular numbers
• 3 Sum of triangular numbers
• 4 Sum of triangular Numbers
• 4 Article: Sums of Squares and Sums of Cubes
• 4 Beautiful Relation: sums of powers
• 4 Units digits of squared numbers
• 5 Numbers expressible as the sum of 2 squares in more than one way
• 5 Product of three consecutive integers not square
• 4 Fibonacci
• 5 Are consecutive Fibonacci numbers coprime?
• 5 The Fibonacci Sequence
• 5 Fibonnaci property
• 5 Fibonacci and Tribonacci sequences
• 5 Divisibility of Fibonacci Numbers
• 5 Proof of Catalan's identity
• 5 Proof of Gelin-Cesaro Identity
• 5 Arithmetic Progressions of Squares
• 5 Squares and cubes problem
• 5 Sum of Square Roots
• 5 4^x + 4^y + 4^z a perfect square
• 5 Perfect Numbers
• 5 f(0)=0 and f(1)=2 => f(7) not square
• 5 Article: More Sums of Squares
• 5 Article: Recent developments on S.P. Numbers
• 6 Primes, powers, and the density of square-free numbers

Prime numbers

• 2 Is 1 a prime number?
• 2 Prime Numbers: prime+prime=prime?
• 3 Composite Numbers
• 3 Prime factorisation
• 4 Euler's phi function - coursework
• 4 Euler's phi function and the Chinese remainder theorem
• 5 Euler's phi function question
• 4 How Many Prime Numbers are there? What is the ratio of picking a prime number?
• 4 Testing whether a large number is prime
• 4 Prime numbers of the form bⁿ+1
• 5 Are there infinite numbers with four or six factors?
• 4 Relatively prime
• 4 The Cayley Purser Algorithm and RSA
• 4 No Primes such that a³+b³=c³
• 5 Factoring large primes and cracking RSA!
• 5 Composite number expression
• 5 When does aⁿ-1 give a prime number?
• 5 Mersenne primes and Germain primes
• 5 Bertrand's conjecture: there is always a prime between n and 2n
• 5 Prime factorisation of n! as n®¥
• 5 Primes/composites of the form 111...11 (repunits)
• 5 Prime Number Program
• 5 Pythagorean Triples -- 2 primes and a square
• 5 Square root of a prime not rational
• 6 Primes, powers, and the density of square-free numbers
• 6 Primes of the form 4n+/-1
• 5 Infinitely many Primes of form 6n+5
• 6 Prime conjecture
• 6 A theorem concerning primes
• 6 The distribution of {px} for all primes p

Infinity and limits

• 3 Biggest number in the world
• 3 Parallel lines and infinity
• 4 Article: Zooming in on the Squares (limits)
• 4 Can you calculate infinity?
• 4 The number infinity
• 4 Does 0.9 recurring equal 1?
• 4 0.99999.... = 1
• 5 Series Summing Methods
• 5 Need conceptual help with -1⁰...
• 5 0⁰
• 5 Paradoxes with infinity
• 5 Countability
• 5 Set of finite strings of 0s and 1s is countable
• 5 Series Summation
• 6 Ramanujan's Square Roots
• 6 No form of Induction for Real Numbers
• 6 x/logx --> ? as x --> 0
• For more on limits, see under Algebra
• For more paradoxes, see separate index

Irrational numbers

• 2 Radicals
• 3 Calculation of p
• 4 The formula for pi
• 3 Why calculate millions of decimal places of pi?
• 4 Exact pi?
• 6 Irrationals
• 5 Relationship between geometrical and series definitions for pi
• 5 Pi's value as an infinite series
• 4 Fraction series tending to p/2
• 5 Proof of a Series for Pi
• 4 Finding square roots
• 4 Surds
• 5 Surds - rationalising the denominator
• 4 The golden number
• 4 Article: Rational and Irrational Numbers
• 4 A proof that all square roots of primes are irrational
• 5 Irrationality of roots of primes
• 5 Proving things irrational
• 5 Proving the n^th root of X is irrational
• 5 Definition of y^x for irrational x
• 5 p^e or e^p
• 5 p irrational
• 5 New Series for Pi
• 6 Series for pi?
• 5 Pi Investaigation
• 5 Irrationality of pi
• 6 Definitions of e
• 6 Euler's Number
• 5 What do irrational powers mean?
• 5 (sqrt2)^(sqrt2)^(sqrt2)
• 6 Rationals and irrationals in any interval
• 6 sqrt(2)^sqrt(2): rational or irrational?
• 5 Ö(6+Ö(6+Ö(6+...
• 6 Computable numbers: Turing machines and pi
• 6 Transcendental Numbers 1
• 6 Transcendental Numbers 2
• 6 Liouville's proof of existence of transcendental numbers
• 5 What are radicals?

Advanced number theory

• 4 Prime value of n⁴+4, square value of 2⁸+2¹¹+2ⁿ
• 5 Is (2n)!/(n!(n+1)!) an integer?
• 5 Given 2ⁿ+1 is an odd prime, prove n=2^k
• 5 (m^b)^a=m(mod n)
• 5 Sigma function is multiplicative
• 5 Perfect Numbers
• 5 Second Perfect number
• 5 Integer equation
• 5 x²+y²+z²=2xyz
• 5 Cantor Set
• 6 Ramsey theory
• 6 Harmonic Sets
• 6 Fermat's Little Theorem
• 6 Wilson's theorem
• 5 Fermat's Little Theorem and Wilson's Theorem
• 5 A novel proof of Fermat's little theorem
• 5 Fermat's Last Theorem
• 5 Algebraic Numbers
• 6 Dirichilet's Theorem
• 6 Modular Forms
• 6 Diophantine equation
• 6 Solving Linear Diophantine Equations
• 6 Does n|f((2ⁿ)-1)?
• 6 f Function
• 6 Quadratic Residues
• 6 Product of consecutive numbers as a power
• 6 Primitive Roots 1
• 6 Primitive roots 2
• 6 P-adic iteration

Miscellaneous

• 2 Article: The Codabar Check
• 2 Ducks, chicken and fish
• 3 Adding Grains of Rice
• 3 Indices - what happens if the index is 0?
• 3 Composite Numbers
• 3 Workers of different speeds
• 4 Laws of indices
• 4 Solutions of a!b!=c!
• 4 Unique expression of integers in a particular form
• 4 Fermat's Last Theorem
• 5 Sum of Consecutive Integers not a Power of 2
• 5 Binomial Theorem to find 1.1⁶
• 5 Fermat's Last Theorem for n=3
• 5 Countability
• 5 ABA/CDC =0.DEFGDEFG...
• 5 sqrt(1+1sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+...
• 5 Cardinal & Ordinal Numbers...?
• 5 Finding square roots by hand
• 5 Finding cube roots by hand
• 5 IMO Inequality Problem
• 5 Miscellaneous problems (unsorted)
• 5 Miscellaneous problems (sorted)
• 5 3 Number Theory Problems
• 5 Another 3 Number Theory Problems
• 5 6 Number Theory Problems
• 5 5 More Number Theory Problems
• 5 Assorted Problems
• 5 Seven Problems
• 5 Number of Ways of Summing to n
• 5 Mathematical constants
• 5 Set Problem
• 5 What are radicals?
• 5 Encryption
• 6 Cantor's Diagonal Argument (real numbers uncountable)
• 6 Uncountably many Countable Sets
• 6 Countability and Cardinality
• 6 Infinity, cardinality, and some relativity
• 6 Schroeder-Bernstein Theorem
• 6 Infinitely Embedded Pattern (Thue Morse constant)
• 6 Grassmann numbers
• 6 Bernoulli Numbers
• 6 The Ultimate Dimension
• 6 Catalan's Constant