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Eratosthenes created a whole section devoted to the examination of Homer, and acquired original works of great tragic dramas of Aeschylus, Sophocles and Euripides. [6] Eratosthenes made several important contributions to mathematics and science, and was a friend of Archimedes. Around 255 BC, he invented the armillary sphere.
An example of "geometric algebra" is: given a triangle (or rectangle, etc.) with a certain area and also given the length of some of its sides (or some other properties), find the length of the remaining side (and justify/prove the answer with geometry). Solving such a problem is often equivalent to finding the roots of a polynomial.
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In this example the fact that the Legendre identity is derived from the Sieve of Eratosthenes is clear: the first term is the number of integers below X, the second term removes the multiples of all primes, the third term adds back the multiples of two primes (which were miscounted by being "crossed out twice") but also adds back the multiples ...
The sieve of Eratosthenes is a popular way to benchmark computer performance. [14] The time complexity of calculating all primes below n in the random access machine model is O(n log log n) operations, a direct consequence of the fact that the prime harmonic series asymptotically approaches log log n.
Image credits: Flares117 Another interesting fact about the family who couldn't sleep comes from Redditor u/Potatoe_expert.Science writer and author of the book about said family The Family That ...
But the fact is, between obscure pieces of information, folklore that has morphed into fact, and even specific details that are hard to believe, true or false questions can be truly hard to figure ...
The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the ...