no one mind this it's just a pre-calc exemption paper thing
Jessica Cantelmo 6/6/05
Mr.Gilbert Pre-Calc Per.4 Research Paper
Prime Numbers: The ancient Greeks mathematicians were the first to study prime numbers and their properties. Mathematicians between the years 500 BC to 300 BC were interested in numbers for their mystical and numerological properties. They were interested in perfect numbers (one whose proper divisors sum to the number itself) and pairs of amicable numbers (a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa). In about 300 BC, Euclid proved in his book, Book XI of the Elements, that there are infinitely many prime numbers. Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers. In the beginning the 17th century, Fermat proved Albert Girard’s speculation that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares. He was able to show how any number could be written as a sum of four squares. Fermat made a new method of factorizing large numbers. Fermat’s Little Theorem states that if p is a prime then for any integer a we have ap = a modulo p. This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n - 2 is divisible by n. The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31 11 is composite. Fermat’s Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.
Euclid of Alexandria was born in about 325 BC and died in about 265 BC in Alexandria, Egypt. Euclid put together the “Elements”. He is the most prominent mathematician of ancient times and was best known for his paper on mathematics. His book was a compilation of knowledge that became the center of mathematical teaching for 2000 years. Euclid showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. There is little known about Euclid of Alexandria’s life except that he taught in Alexandria.
Leonhard Euler was born April 15th, 1707 in Basel, Switzerland and died September 18th 1783 in St. Petersburg, Russia. He was taught elementary mathematics from his father Paul Euler while growing up and his interest in mathematics was sparked when he was taught the elementary mathematics from his father. He entered the University of Basel in 1720 at the age of 14. He did this so he could obtain a general education before going on to more advanced studies. His father wanted him to be a devout Christian but Leonhard had no desire or enthusiasm for the study of theology so he persuaded his father into allowing him to switch majors so he could study mathematics instead. Euler left Basel in 1727 when he knew he was not going to be appointed the chair of physics. He went to the Rhine and joined the St. Petersburg Academy of Sciences. He was then appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post. Euler's health problems began in 1735 when he had a severe fever and almost lost his life. His eyesight problems began in 1738 with overstrain due to his cartographic work.
Pierre de Fermat was born August 17th 1601 in Beaumont-de-Lomagne, France and died on January 16th 1665 in Castres, France because of a plague that struck the region in the early 1650s. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius’s Plane loci to one of the mathematicians there. From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court.
Mr.Gilbert Pre-Calc Per.4 Research Paper
Prime Numbers: The ancient Greeks mathematicians were the first to study prime numbers and their properties. Mathematicians between the years 500 BC to 300 BC were interested in numbers for their mystical and numerological properties. They were interested in perfect numbers (one whose proper divisors sum to the number itself) and pairs of amicable numbers (a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa). In about 300 BC, Euclid proved in his book, Book XI of the Elements, that there are infinitely many prime numbers. Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers. In the beginning the 17th century, Fermat proved Albert Girard’s speculation that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares. He was able to show how any number could be written as a sum of four squares. Fermat made a new method of factorizing large numbers. Fermat’s Little Theorem states that if p is a prime then for any integer a we have ap = a modulo p. This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n - 2 is divisible by n. The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31 11 is composite. Fermat’s Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.
Euclid of Alexandria was born in about 325 BC and died in about 265 BC in Alexandria, Egypt. Euclid put together the “Elements”. He is the most prominent mathematician of ancient times and was best known for his paper on mathematics. His book was a compilation of knowledge that became the center of mathematical teaching for 2000 years. Euclid showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. There is little known about Euclid of Alexandria’s life except that he taught in Alexandria.
Leonhard Euler was born April 15th, 1707 in Basel, Switzerland and died September 18th 1783 in St. Petersburg, Russia. He was taught elementary mathematics from his father Paul Euler while growing up and his interest in mathematics was sparked when he was taught the elementary mathematics from his father. He entered the University of Basel in 1720 at the age of 14. He did this so he could obtain a general education before going on to more advanced studies. His father wanted him to be a devout Christian but Leonhard had no desire or enthusiasm for the study of theology so he persuaded his father into allowing him to switch majors so he could study mathematics instead. Euler left Basel in 1727 when he knew he was not going to be appointed the chair of physics. He went to the Rhine and joined the St. Petersburg Academy of Sciences. He was then appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post. Euler's health problems began in 1735 when he had a severe fever and almost lost his life. His eyesight problems began in 1738 with overstrain due to his cartographic work.
Pierre de Fermat was born August 17th 1601 in Beaumont-de-Lomagne, France and died on January 16th 1665 in Castres, France because of a plague that struck the region in the early 1650s. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius’s Plane loci to one of the mathematicians there. From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court.