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Transcript
FAMOUS CONJECTURES TOP FIVE 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. 5. FOUR COLOR THEOREM STATEMENT 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. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions Example 4. LEGENDRE’S CONJECTURE Examples STATEMENT 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 9 and 16 are 11 and 13 3. CONJECTURE TWIN PRIME NUMBERS Examples STATEMENT There are infinitely many p=3 and p+2 = 5 primes p such that p+2 is also p=5 and p+2 = 7 prime. p = 11 and p+2 = 13 p = 29 and p+2 = 31 2 . G O L D BA C H ’ S C O N J E C T U R E STATEMENT Examples Every even integer greater 4 = 2+2 than 2 can be expressed as 6 = 3+3 the sum of two primes. 8 = 3+5 10 = 3+7 = 5+5 1. FERMAT’S LAST THEOREM Example STATEMENT There are no positive integers a, For n=2 b and c, can satisfy the equation a=3 an + bn = c n for any integer value of n greater than two. b=4 then 32 + 42 = 52 c=5 « 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