Simple Proofs
Simple proofs often involve simple manipulation of formulae or expressions. Proving an identity is often quite simple, and often shows several examples as an aid. For example
The sum of any four consecutive numbers is equal to the product of the largest two minus the product of the smallest two.
and
and
and
A general proof however requires algebra, and the choice of four general consecutive numbers,
and
These expressions are the same so the proof is complete.
Abstract proofs require more though. Suppose we want to prove that the product of consecutive integers is even.
All these are even, but as before, for a general proof we require general consecutive numbers,
and
Ifis odd then
is even and so
is even.
Ifis even then multiplying by any number is even, sois even.
Often it is required to disprove a statement by finding a counterexample.
The expression
returns a prime number for
right up to
etc
For
A more obvious number to choose is
since if
and 11 are all divisible by 11. In fact, no formula of the form
generates only prime number. If you choose
you will always get a composite (non – prime) number.