corresponding come a construction of point out which type a square pyramid, is referred to as a square pyramidal number (or sometimes, just a pyramidal number). The first few are 1, 5, 14, 30, 55, 91, 140, 204, ... (OEIS A000330). The generating role for square pyramidal number is


*

*

where

*
is the
*
th triangle number.

You are watching: Which of the following is the pyramidal number sequence

The only numbers i m sorry are at the same time square

*
and square pyramidal
*
(the cannonball problem) room
*
and
*
, equivalent to
*
and also
*
(Ball and Coxeter 1987, p.59; Ogilvy 1988; Dickson 2005, p.25), together conjectured through Lucas (1875), partially proved by Moret-Blanc (1876) and also Lucas (1877), and proved by Watson (1918). The difficulty requires solving the Diophantine equation


*

(Guy 1994, p.147). Watson (1918) gave an practically elementary proof, disposing of most instances by elementary school means, yet resorting come the use of elliptic features for one pesky case. Completely elementary proofs have actually been offered by Ma (1985) and also Anglin (1990).

Numbers i m sorry are simultaneously triangular

*
and also square pyramidal
*
satisfy the Diophantine equation


*

*

The just solutions room

*
, (0, 0), (1, 1), (5, 10), (6, 13), and also (85, 645) (Guy 1994, p.147), equivalent to the nontrivial triangle square pyramidal numbers 1, 55, 91, 208335.

Numbers which are concurrently tetrahedral

*
and square pyramidal
*
fulfill the Diophantine equation


Beukers (1988) has actually studied the problem of finding remedies via integral points on one elliptic curve and also found that the just solution is the trivial

*
.


REFERENCES:

Anglin, W.S. "The Square Pyramid Puzzle." Amer. Math. Monthly 97,120-124, 1990.

Anglin, W.S. The Queen the Mathematics: An arrival to Number Theory. Dordrecht, Netherlands: Kluwer, 1995.

Baker, A. And also Davenport, H. "The Equations

*
and also
*
." Quart J. Math. Ser. 2 20, 129-137, 1969.

Ball, W.W.R. And Coxeter, H.S.M. MathematicalRecreations and also Essays, 13th ed. Brand-new York: Dover, p.59, 1987.

Beukers, F. "On Oranges and Integral clues on certain Plane Cubic Curves."Nieuw Arch. Wisk. 6, 203-210, 1988.

Conway, J.H. And Guy, R.K. TheBook the Numbers. Brand-new York: Springer-Verlag, pp.47-50, 1996.

Dickson, L.E. History of the theory of Numbers, Vol.2: Diophantine Analysis. New York: Dover, 2005.

Guy, R.K. "Figurate Numbers." §D3 in Unsolved problems in Number Theory, second ed. New York: Springer-Verlag, pp.147-150, 1994.

Kanagasabapathy, P. And Ponnudurai, T. "The simultaneous Diophantine Equations

*
and
*
." Quart. J. Math. Ser. 2 26, 275-278, 1975.

Ljunggren, W. "New systems of a trouble Posed through E.Lucas." NordiskMat. Tidskrift 34, 65-72, 1952.

Lucas, É. Inquiry 1180. Nouv. Ann. Math. Ser. 2 14, 336, 1875.

Lucas, É. Equipment de concern 1180. Nouv. Ann. Math. Ser. 2 15,429-432, 1877.

Ma, D.G. "An Elementary proof of the systems to the Diophantine Equation

*
." Sichuan Daxue Xuebao 4, 107-116, 1985.

Moret-Blanc, M. Question 1180. Nouv. Ann. Math. Ser. 2 15, 46-48, 1876.

Ogilvy, C.S. And also Anderson, J.T. Excursionsin Number Theory. New York: Dover, pp.77 and 152, 1988.

Sloane, N.J.A. Succession A000330/M3844in "The On-Line Encyclopedia of integer Sequences."

Watson, G.N. "The problem of the Square Pyramid." Messenger. Math. 48,1-22, 1918.

Wolf, T. "The

*
Puzzle." http://home.tiscalinet.ch/t_wolf/tw/misc/squares.html.


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