[math-fun] The "square pyramid" puzzle
This post concerns the uniqueness of the nontrivial solution to (*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2 (namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990. The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base." (He was evidently not counting the trivial solution of n = K = 1.) In 1876 a flawed proof was pubished by M. Moret-Blanc. In 1877 a flawed proof was pubished by Lucas. In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions. More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level. In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages). My question to math-fun is this: Can this problem really be as difficult as this history would make it seem? Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof? Or perhaps there could be a geometric argument? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
See D3 of UPINT. There are ``elementary'' proofs, but not quite as elementary as you might hope for. R. On Wed, 16 Apr 2008, Dan Asimov wrote:
This post concerns the uniqueness of the nontrivial solution to
(*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2
(namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990.
The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base."
(He was evidently not counting the trivial solution of n = K = 1.)
In 1876 a flawed proof was pubished by M. Moret-Blanc.
In 1877 a flawed proof was pubished by Lucas.
In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions.
More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level.
In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages).
My question to math-fun is this: Can this problem really be as difficult as this history would make it seem?
Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof?
Or perhaps there could be a geometric argument?
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This problem has a special meaning for me. In my thesiss(in a part written in 1973) I proved a result which provided a simple elementary method of finding all integer points on an elliptic curve (of which the square pyramid problem is an example) provided that there was a surjective isogeny from another elliptic curve onto the rational points (this surjectivity is determined by a descent calculation, which can be phrased in elementary terms). John Coates asked if my method could apply to the "square pyramid" problem. Alas, I couldn't get it to work. I did spend a lot of time studying Watson's paper, and came to the conclusion that it was some sort of p-padic method in disguise, but I could never penetrate all the details. Victor On Wed, Apr 16, 2008 at 5:54 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This post concerns the uniqueness of the nontrivial solution to
(*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2
(namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990.
The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base."
(He was evidently not counting the trivial solution of n = K = 1.)
In 1876 a flawed proof was pubished by M. Moret-Blanc.
In 1877 a flawed proof was pubished by Lucas.
In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions.
More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level.
In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages).
My question to math-fun is this: Can this problem really be as difficult as this history would make it seem?
Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof?
Or perhaps there could be a geometric argument?
--Dan
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
For those interested, I append, below the copied message, the present state of part of D3 of UPINT. Comments are welcome and may be incorporated in the next edition. R. On Wed, 16 Apr 2008, victor miller wrote:
This problem has a special meaning for me. In my thesiss(in a part written in 1973) I proved a result which provided a simple elementary method of finding all integer points on an elliptic curve (of which the square pyramid problem is an example) provided that there was a surjective isogeny from another elliptic curve onto the rational points (this surjectivity is determined by a descent calculation, which can be phrased in elementary terms). John Coates asked if my method could apply to the "square pyramid" problem. Alas, I couldn't get it to work. I did spend a lot of time studying Watson's paper, and came to the conclusion that it was some sort of p-padic method in disguise, but I could never penetrate all the details.
Victor
On Wed, Apr 16, 2008 at 5:54 PM, Dan Asimov <dasimov@earthlink.net> wrote:
This post concerns the uniqueness of the nontrivial solution to
(*) 1^2 + 2^2 + 3^3 + . . . + n^2 = K^2
(namely, n = 24 and K = 70), and is largely based on "The square pyramid puzzle" by W.S. Anglin, American Mathematical Monthly, Feb. 1990.
The problem seems to have originated with an 1875 challenge from Edouard Lucas to prove that "A square pyramid of cannonballs contains a square number of cannonballs only when it has 24 cannonballs along its base."
(He was evidently not counting the trivial solution of n = K = 1.)
In 1876 a flawed proof was pubished by M. Moret-Blanc.
In 1877 a flawed proof was pubished by Lucas.
In 1918 -- 43 years after the problem was posed -- apparently the first valid proof (14 pages) was published, by G.N. Watson, using an extended theory of Jacobi elliptic functions.
More proofs appeared in 1952, 1966, and 1975, but not until 1985 did someone (De Gang Ma) publish a proof accessible at the undergraduate level.
In a simplification of De Gang Ma's method, in this article Anglin uses 3 lemmas to cover the case of n even, and 7 lemmas (and an entirely different strategy) to cover the case of n odd -- and draws on the theory of Pell's equation and quadratic reciprocity. The arguments involve considering a large number of cases and strike me as quite ad hoc, (although in total they cover only about 3 1/2 pages).
My question to math-fun is this: Can this problem really be as difficult as this history would make it seem?
Is it possible that considering the equation (*) modulo just the right primes would lead to an efficient proof?
Or perhaps there could be a geometric argument?
--Dan
[except from D3 in UPINT} The ``triangle = tetrahedron'' problem is a special case of a more general question about equality of binomial coefficients (see {\bf B31}) --- the only nontrivial examples of ${n\choose2}={m\choose3}$ are $(m,n)$ = (10,16), (22,56) and (36,120) and de Weger confirms that there are no nontrivial examples of ${n\choose2}={m\choose4}$ apart from (10,21). The case ``square pyramid = square'' is Lucas's problem. Is $x=24$, $y=70$ the only nontrivial solution of the diophantine equation $$y^2=x(x+1)(2x+1)/6\ ?$$ This was solved affirmatively by Watson, using elliptic functions, and by Ljunggren, using a Bhaskara equation (often called a Pell equation) in a quadratic field. Mordell asked if there was an elementary proof, and affirmative answers have been given by Ma, by Xu \& Cao, by Anglin and by Pint\'er. The same equation in disguise is to ask if (48, 140) is the unique nontrivial solution to the case ``square = tetrahedron'', since the previous equation may be written $$(2y)^2=2x(2x+1)(2x+2)/6,$$ though, as Peter Montgomery notes, this doesn't eliminate the possibility of an odd square. A more modern treatment is to put $12x=X-6$, $72y=Y$ and note that $Y^2=X^3-36X$ is curve 576H2 in John Cremona's tables. The point (12,36) (which gives an odd square) serves as a generator. There's an infinity of rational solutions, but the only nontrivial integer solution to the original problem is given by the point $(294,5040)$. ----------------
participants (3)
-
Dan Asimov -
Richard Guy -
victor miller