“Diophantine Equation--Linear”的概念、定义、习题解题方法及翻译-数学辞典

单词

Diophantine Equation--Linear

释义

Diophantine Equation--Linear

A linear Diophantine equation (in two variables) is an equation of the general form

(1)

where solutions are sought with

, and

Integers. Such equations can be solved completely, andthe first known solution was constructed by Brahmagupta. Consider the equation

(2)

Now use a variation of the Euclidean Algorithm, letting

and

			(3)
			(4)
			(5)
			(6)

Starting from the bottom gives

			(7)
			(8)


			(9)

Continue this procedure all the way back to the top.

Take as an example the equation

(10)

Proceed as follows.

The solution is therefore

. The above procedure can be simplified by noting that the two left-mostcolumns are offset by one entry and alternate signs, as they must since

			(11)
			(12)
			(13)

so the Coefficients of

and

are the same and

(14)

Repeating the above example using this information therefore gives

and we recover the above solution.

Call the solutions to

(15)

and

. If the signs in front of

are Negative, then solve the above equation and take the signs ofthe solutions from the following table:

equation

In fact, the solution to the equation

(16)

is equivalent to finding the Continued Fraction for

, with

and

Relatively Prime (Olds 1963).If there are

terms in the fraction, take the

th convergent

. But

(17)

so one solution is

, with a general solution

			(18)
			(19)

with

an arbitrary Integer. The solution in terms of smallest Positive Integers is givenby choosing an appropriate

Now consider the general first-order equation of the form

(20)

The Greatest Common Divisor

can be divided through yielding

(21)

where

, and

. If

, then

is not an Integer and theequation cannot have a solution in Integers. A necessary and sufficient condition for the generalfirst-order equation to have solutions in Integers is therefore that

. If this is the case, thensolve

(22)

and multiply the solutions by

, since

(23)

References

Courant, R. and Robbins, H. ``Continued Fractions. Diophantine Equations.'' §2.4 in Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 49-51, 1996.

Dickson, L. E. ``Linear Diophantine Equations and Congruences.'' Ch. 2 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 41-99, 1952.

Olds, C. D. Ch. 2 in Continued Fractions. New York: Random House, 1963.

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