This document lists and describes 5 famous unproven conjectures in mathematics: 1) Fermat's Last Theorem, which states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer n greater than two. 2) Goldbach's Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two primes. 3) The Twin Prime Conjecture, which states that there are infinitely many prime numbers p such that p+2 is also prime. 4) Legendre's Conjecture, which states that there is a prime number between n2 and (n + 1)2 for every positive integer n

1. Famous conjectures TOP FIVE KAREN LOPEZ B.

2. A conjecture is a proposition that is unproven but appears correct and has not been disproven. After demostrating the truth of a conjecture, this came to be considered a theorem and as such can be used to build other formal proofs.

3. Given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. 5. FOUR COLOR THEOREM STATEMENT Example

4. There is a prime number between n2 and (n + 1)2 for every positive integer n. n=1 Between 1 and 4 are 2 and 3 n=2 Between 4 and 9 are 5 and 7 n=3 Between 9and 16 are 11 and 13 4. LEGENDRE’S CONJECTURE STATEMENT Examples

5. There are infinitely many primes p such that p+2 is also prime. 3. Conjecture twin prime numbers STATEMENT Examples p = 3 and p+2 = 5 p = 5 and p+2 = 7 p = 11 and p+2 = 13 p = 29 and p+2 = 31

6. Every even integer greater than 2 can be expressed as the sum of two primes. 4 = 2+2 6 = 3+3 8 = 3+5 10 = 3+7 = 5+5 2. Goldbach’s Conjecture Examples STATEMENT

7. There aren’t three positive integers a, b and c, can satisfy the equation an + bn = cn for any integer value of n greater than two. For n=2 a=3 b=4 c=5 then 32 + 42 = 52 1. fermat’s last theorem STATEMENT Example

8. « I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it. » Pierre de Fermat[, 1637