| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
[ This problem was proposed by Donald Cross, as problem 1129,
in Vol 12, No 3 of Crux Mathematicorum ]
(a) Show that every positive whole number >= 84 can be written as the sum
of three positive whole numbers in at least four ways (all twelve
numbers different) such that the sum of the squares of the three
numbers in any group is equal to the sum of the squares of the three
numbers in each of the other groups.
(b) Same as part (a), but with "three" replaced by "four" and "twelve"
replaced by "sixteen".
(c) Is 84 minimal in (a) and/or (b)?
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 487.1 | Some related info, and one more problem. | THEBUS::KOSTAS | Kostas G. Gavrielidis <o.o> | Mon May 12 1986 23:41 | 42 |
re. .0
Here are some theroms which may help
Theorem 1.0 A positive integer n can be expressed as the sum of
two squares if and only if every prime factor of n of the form
4k + 3 occurs with even multiplicity.
As a corollary, we note that if n has no prime factor congruent
to 3(mod 4) then n is the sum of two squares, In applying
the theorem, we allow 0^2 as one of the terms, and we also
allow identical terms.
Theorem 1.1 Every positive integer is the sum of four squares.
According to this theorem the Diophantine equation
2 2 2 2
v + w + x + y = z
has solution for every positive integer z.
A proof is in Niven, Ivan and H. S. Zuckerman. An Introduction
to the Theory of Numbers, 3rd ed. New York: John Wiley & Sons, Inc.,
1972.
Theorem 1.2 Every positive integer except 1, 2, 3, 4, 6, 7, 9,
10, 12, 15, 18, and 33 is the sum of five (not necessarily distinct)
positive squares.
Also another problem:
Find the least positive integer which cannot be expressed as the
sum of fewer that nine cubes.
Enjoy,
Kostas G.
| |||||
| 487.2 | CLT::GILBERT | Juggler of Noterdom | Tue May 13 1986 03:21 | 14 | |
re .1: It's really easier than all that. re .0: (c) 84 is minimal in (a). 64 is minimal in (b). Proving that these are minimal is exhausting. In fact (for 64, and 4 squares): 31^2 + 13^2 + 12^2 + 8^2 = 29^2 + 19^2 + 10^2 + 6^2 = 28^2 + 17^2 + 16^2 + 3^2 = 25^2 + 24^2 + 11^2 + 4^2 = 23^2 + 22^2 + 18^2 + 1^2 | |||||
| 487.3 | some history and a solution | THEBUS::KOSTAS | Kostas G. Gavrielidis <o.o> | Wed Jun 04 1986 22:20 | 30 |
re. .1
Theorem 1.1 which states: "Every positive integer is the sum of
four squares", was assumed by Diophantus and was proven by Lagrange
(1770).
It is also natural to attempt a generalization of this theorem to
higher power. Edward Waring (1734-1798) stated without proof that
every positive integer can be expressed as the sum of nine cubes
and as the sum of 19 fourth powers.
Also the solution to:
Find the least positive integer which cannot be expressed as
the sum of fewer than nine cubs,
is
23
kgg
| |||||
| 487.4 | CLT::GILBERT | eager like a child | Tue Apr 21 1987 12:48 | 1 | |
No proof has yet been posted here. | |||||