Diophantine Equations
Diophantus, often called the father of Algebra, was a 3rd Century Greek mathematician( he lived around 250A.D) who is credited for writing one of the first books on algebra "Arithmetica" and was one of the first mathematicians to recognize fractions as positive integers. Very little about his life is known than that what was written down on his tombstone: God granted him to be a boy for the sixth part of his life, How old was Diophantus when he died? This riddle is an example for the kind of problems in which Diophantus was interested in. |
Diophantine equations are indeterminate polynomial equations in which only integer solutions are allowed. (1)For a 1st order Diophantine equation, there is an algorithm to determine whether there exists a solution. A linear Diophantine equation of 2 variables is of the form Ax + By = C where A, B and C are integers. The first known solution was furnished by Brahmagupta. (2)Fibonacci numbers have a diophantine representation. A pair of consecutive fibonacci numbers do indeed satisfy the equation: x2+xy-y2 = ±1 and the solutions (x, y) in positive integers to the equation (x2+xy-y2)2 - 1= 0 are the pairs of consecutive Fibonacci numbers. (3) The linear Diophantine equation ax-by=c with integers a, b, c is solvable if and only if gcd of a and b divides c OR (b, a) |c. Example- take 38x-15y=1. Various solutions (x, y) are (-13,-33),(2,5),(17,43). solutions yield good approximations to 15/38 i.e. (x/y) --> 15/38.One can work out on other equn. 3x-5y=19, 617x-125y=91 etc. If the linear Diophantine equation is solvable, there are infinite number of integer solutions. (4) Euclid showed that if there exist 2 integers a, b , then gcd (a, b) can be expressed as a linear combination of a and b. Or ax + by = gcd (a, b) |
Some Observations (1) If Ax -By=C is a Diophantine equation, and a solution is found out at x=m, y=n, then (m+ B, n+ A),(m+2B,n+2A),(m+3B,n+3A,...(m+ kB, n+ kA) are also solutions where k is an integer (positive or negative). (2) If Ax -By=C is a diophantine equation & the lowest difference between x, y i.e. x-y =Δ , then in general x-y = Δ +k (B-A) where k is an integer. (3) If Ax -By=C is a Diophantine equation & the lowest sum between x, y i.e. x+ y =δ , then in general x+ y =δ + k (B+ A) where k is an integer, positive or negative. (4) If Ax -By = C is a Diophantine equation, no. of lowest values that y can take is A irrespective of the value of C and B. For example, if 2x-5y=C, the lowest value of y can be 1 or 2--> only these 2 values whether C is 1,2,3,4,17 etc and /or B is of any value. Again , if 3x-5y=C, lowest y can take only any of the 3 values (1,2 or 3). If 6x-5y=C, y shall take any of 6 values (1,2,3,4,5,6) irrespective of value of B, C. (5) If Ax -By = C is a Diophantine equation where y= m, then y = m for the equation Ax -By = C ± kA where k is an integer. Example -- In the equation, 15x-7y =4 , y=8 (lowest value). Then y = 8 (lowest value) also for the equation 15x-7y =19 and 15x-7y=34 etc. (6) If Ax -By = C is a Diophantine equation for which lowest value of y is m, then for Ax-(B+kA)y = C, y=m where k is an integer. (7) If Ax-By = C is a diophantine equation & lowest value of y is m, then for kAx-kBy=kC, lowest value of y shall also be m where k is an integer. |