New computer methods reveal secrets of ancient math problem
According to Brian Conrey, Director of the American Institute of Mathematics, “old problems like this may seem obscure, but they generate a lot of interesting research and useful as people develop new ways to attack.”
The problem was first posed more than a thousand years ago, refers to the areas of triangles. The problem is surprisingly difficult to determine which integers can be the area of a triangle whose sides are integers or fractions. The area of triangle is called a number “congruent”. For example, the 3-4-5 right triangle that students are in geometry has an area of 1 / 2 x 3 x 4 = 6, so 6 is a congruent number. The smallest number congruent is 5, which is the area of triangle with sides 3 / 2, 20 / 3, and 41 / 6. The first numbers are consistent about 5, 6, 7, 13, 14, 15, 20 and 21. Many congruent numbers were known before the recalculation. For example, each number in the sequence of 5, 13, 21, 29, 37, …, is a congruent number. However, other sequences of similar appearance, such as 3, 11, 19, 27, 35, …., are more mysterious and each number has to be reviewed individually.
The calculation given new numbers 3148379694 congruent to one trillion dollars.
Consequences, and future plans
Team members Bill Hart said: “The hard part was developing a quick general library computer codes for such calculations. Once we had not spent much time to write the specialized software needed for this calculation in particular. ” The software used in the calculation is freely available, and anyone with a larger computer can be used to break the team record or make other similar calculations.
Besides the practical advances necessary for this result, the response also has theoretical implications. As the mathematician Michael Rubinstein, University of Waterloo, “Some years ago, we combine the ideas of number theory and physics to predict how they behave statistically consistent numbers. I am very pleased to see that our prediction was quite accurate. ” It was Rubinstein who challenged the team to attempt this calculation. Rubinstein method predicts that about 800 billion more consistent number to a quadrillion, a prediction that could determine whether the equipment with a hard drive big enough were available.
History of problem
The congruent number problem was reported by the Persian mathematician Al-Karaji (c.953 – c.1029). His version involves not triangles, but is expressed in terms of square numbers, numbers that are squares of integers: 1, 4, 9, 16, 25, 36, 49, …, or squares of rational numbers : 25 / 9, 49/100, 144/25, etc. Question: What integers n Is there a square-A2-a2 n a2 + n are squares? When this occurs, n is called a congruent number. The name comes from the fact that there are three positions that are congruent modulo n. A major influence on al-Karaji was the translation into Arabic of works of the Greek mathematician Diophantus (C.210 – c.290), which raises similar problems.
A small amount of progress in the next thousand years. In 1225, Fibonacci (for “Fibonacci numbers” fame) showed that the numbers 5 and 7 were consistent, and declared, but not prove, that the 1 is not a congruent number. The test was provided by Fermat (of “Fermat’s last theorem” fame) in 1659. In 1915 the congruent numbers less than 100 had been determined, and in 1952 introduced Kurt Heegner deep mathematical techniques in the field and has shown that all prime numbers in the sequence of 5, 13, 21, 29, …, are consistent . But in 1980 there were still 1,000 cases of children who had not been resolved.
Jerrold Tunnell 1982 at Rutgers University made significant progress by exploiting the connection (first use Heegner) between the congruent numbers and elliptic curves, mathematical objects for which there is a well-established theory. He found a simple formula to determine whether a number is a congruent number. This allowed the first thousand cases that were resolved very quickly. One issue is that the full realization of their formula depends on the truth of a particular case of one of the outstanding problems in mathematics known as the Birch and Swinnerton-Dyer Conjecture. That conjecture is one of seven Millennium Prize Problems posed by the Clay Mathematics Institute with a prize of one million dollars.
Results like these are sometimes skeptical because of the complexity of carrying out this calculation and the possibility of large errors in the computer or programming. The researchers took special care to verify their results, making the calculation twice, on different computers, using different algorithms, written by two independent groups. The team of Bill Hart (University of Warwick, in England) and Tornari Gonzalo (Universidad de la República, Uruguay) uses the computer “Selmer” at the University of Warwick. Selmer is funded by the Engineering and Physical Sciences Research Council in the UK. Most of its code was written during a workshop at the University of Washington in June 2008.
The team of Mark Watkins (University of Sydney, Australia), David Harvey (Courant Institute, New York University, New York) and Robert Bradshaw (University of Washington in Seattle) used the computer “learned” at the University of Washington . Sage is funded by the National Science Foundation in the U.S.. The computer code was developed during a workshop at the Science Center in Benasque Benasque Pedro Pascual, Spain, in July 2009. Both workshops were supported by the American Institute of Mathematics through a research grant focused group of National Science Foundation.
Source: American Institute of Mathematics
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