Diophantine Analysis

Diophantine Analysis, in mathematics, branch of number theory concerned with determining the solutions in integers of algebraic equations with two or more unknowns. Such problems were treated by Greek mathematicians Pythagoras, Diophantus and others.

Typical problems in Diophantine analysis would be to find two integers such that the sum of their squares is a square (3 and 4, or 5 and 12); or two integers such that the sum of their cubes is a square (1 and 2); or three integers such that their squares are in arithmetic progression (1, 5, and 7). In algebraic terms, the three examples call for integers x, y, z, ..., such that x2 + y2 = z2; x3 + y3 = z2; and x2 + z2 = 2y2, respectively. Usually an attempt is made to determine if a problem has infinitely many or a finite number of solutions, or none. Common examples of problems that involve Diophantine analysis are money budgeting problems, number theories and fixed dimensional designs. Recent approaches to problems of this sort involve the use of high-speed computers to find solutions or to establish counterexamples to the conjectured theory. Manual calculations are usually accomplished with Euclid’s Principle to obtain the possible results for linear equations but analysis is more or less based on scientific trial and error solutions. Even supercomputers have difficulty in analyzing second degree equations that involve finding integral solutions and indeed solutions could take up to powers greater than 50 just to obtain the smallest integral solution to a certain equation. Algorithms are still developed to calculate this type of equations.

Posted 2011-01-24 and updated on Jan 24, 2011 8:07am by crisd

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