Diophantine Technique
Solving a linear equation in two unknowns for integer solutions
In 2004 the cicadas visited the middle Atlantic region. They visit every 17 years. The previous time was 1987. And, wait, Halley's Comet visited about then. It was 1986. It visits every 76 years. When will they both visit in the same year? Allow me to consider the periods as exact years.
A linear equation in 2 unknowns of which
we desire an integer solution
Now to solve this equation, divide the equation by the smaller of the two coefficients
Isolate fractional i on one side of the equation
Note that j - 4*i must be an integer. Let's simplify by giving it its own name
Multiply by 15 - to make the remainder, after division by 17, one
Isolate fractional i on one side of the equation
Note that 15*k1 -7*i must be an integer. Let's simplify by giving it its own name
Multiply by 17 to remove the divisors
i must always be equal to 17 times an integer plus 15 more
