This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

Title: Cliff Puzzle 16: Undulating Squares

From: cliff@watson.ibm.com

If you respond to this puzzle, if possible please send me your name,

address, affiliation, e-mail address, so I can properly credit you if

you provide unique information. PLEASE ALSO directly mail me a copy of

your response in addition to any responding you do in the newsgroup. I

will assume it is OK to describe your answer in any article or

publication I may write in the future, with attribution to you, unless

you state otherwise. Thanks, Cliff Pickover

* * *

A square number is of the form y=x**2. For example, 25 is a square

number.

Undulating numbers are of the form: ababababab... For example, the

following are undulating numbers: 1717171, 282828, etc.

1. Are there any undulating square numbers?

2. Are there any undulating cube numbers?

pickover/pickover.16.s

-------------------------

In article <1992Oct30.175102.142177@watson.ibm.com> you write:

: 1. Are there any undulating square numbers?

11^2 = 121

: 2. Are there any undulating cube numbers?

7^3 = 343

(yes, I know they're short, but they qualify!)

-- Michael Neylon aka Masem the Great and Almighty Thermodynamics GOD! // | Senior, Chemical Engineering, Univ. of Toledo \\ // Only the | Summer Intern, NASA Lewis Research Center \ \X/ AMIGA! | mneylon@jupiter.cse.utoledo.edu / --------+ How do YOU spell 'potato'? How 'bout 'lousy'? +---------- "Me and Spike are big Malcolm 10 supporters." - J.S.,P.L.C.L -------------------------

In article <1992Oct30.204134.97881@watson.ibm.com> you write:

>Hi, I was interested in non-trivial cases. Those with greater

>than 3 digits. Award goes to the person who finds the largest

>undulating square or cube number. Thanks, Cliff

343 and 676 aren't trivial (unlike 121 and 484 it doesn't come from

obvious algebraic identities). The chance that a "random"

number around x should be a perfect square is about 1/sqrt(x);

more generally, x^(-1+1/d) for a perfect d-th power. Since

there are for each k only 90 k-digit undulants you expect

to find only finitely many of these that are perfect powers,

and none that are very large. But provably listing all cases

is probably only barely, if at all, possible by present-day

methods for treating exponential Diophantine equations, unless

(as was shown in a rec.puzzles posting re your puzzles on

arith. prog. of squares with common difference 10^k) there is

some ad-hoc trick available. At any rate the largest undulating

power is probably 69696=264^2, though 211^3=9393931 comes

remarkably close.

--Noam D. Elkies

-------------------------

In article <1992Oct30.175102.142177@watson.ibm.com>, you write...

>1. Are there any undulating square numbers?

>

Other than the obvious 11**2, 22**2, and 26**2, there is 264**2

which equals 69696.

>2. Are there any undulating cube numbers?

>

Just 7**3 as far as I can tell, though I'm limited to IEEE computational

reals.

PauL M SchwartZ (-Z-) | Follow men's eyes as they look to the skies v206gb6c@ubvms.BitNet | the shifting shafts of shining pms@geog.buffalo.edu | weave the fabric of their dreams pms@acsu.buffalo.edu | - RUSH -

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