Examples
The well known Pell equation
is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns precisely when the parameter is 0 or not a perfect square. In the present context, one says that this equation provides a Diophantine definition of the set
- {0,2,3,5,6,7,8,10,...}
consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:
- The equation only has solutions in when a is not a power of 2.
- The equation only has solutions in when a is greater than 1 and is not a prime number.
- The equation defines the set of pairs such that
Read more about this topic: Diophantine Set
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)