Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 79
primality has in most cases required great labor from its
discoverers.
10. MULTIGRADE
A multigrade is a relation between numbers. It is of the
form
:Ea'" = :ElYand
holds good for more than one value of x. The simplest
multigrade is
1"'+2"'+6"'=4"'+5"',
which holds for x = 1 or 2, inasmuch as 1 + 2 + 6 = 4 + 5
and 12 + 22 + 62 = 42 + 52. One can find multigrades which
hold for as many values of x as one pleases. Here is a very
useful theorem.
Given a muUigrade
we can form a new muUigrade,
1 :i % :i P.
tat' + t(b,: + c)" = fbt' + tea. + c)"', 1 :i % :i P + 1,
.-1 i-1 i-I .-1
where c is any constant.
This is a very efficient method of forming multigrades for
arbitrarily high powers. For example, we may start with
the identity 1 + 9 = 4 + 6. If we take c = 1 we have
1'" + 5"+ 7" + 9" = 2" + 4'" t- 6" + 10z, x = 1, 2. Or we may
take c = 2 and get 1"+ 8"+ 9" = 3"+ 4"'+ 11", x = 1, 2.
From the latter, with c = 1, we find 1"'+ 5"'+ 8"'+ 12" =
2"+ 3"'+ 10z+ 11", x = 1, 2, 3. We can continue, letting c
have any succession of values, as c = 7, 4, 8, 1, 13, 11,···.
11. CRYPTARITHMETIC
Under the title "Cryptarithmie" (which we have translated
as "Cryptarithmetic") the journal Sphinx published,