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From: pumphrey@ttidcb.UUCP (Larry Pumphrey)
Newsgroups: net.math
Subject: Re: A real problem! (Not a polar bear problem)
Message-ID: <552@ttidcb.UUCP>
Date: Tue, 19-Nov-85 12:52:25 EST
Article-I.D.: ttidcb.552
Posted: Tue Nov 19 12:52:25 1985
Date-Received: Wed, 20-Nov-85 20:57:12 EST
Organization: TTI, Santa Monica, CA.
Lines: 54


>     Enough silly polar bear problems.
>     Here's a problem that requires a little more sophistication:
>
>         The 3-4-5 triangle has integral area (area=6).
>         The 13-14-15 triangle has integral area (area=84).
>         Find all triangles with sides n-(n+1)-(n+2) that have
>         integral area.
>
>         Note:  This isn't an elementary problem.
>         I'll send the answer to anybody who wants it, and maybe even post
>         it . . .


  Yes, this is quite a problem!  The answer is given below in rot13 format
  but I also won't post the proof as it is rather lengthy.

  Guvf ceboyrz unf na vasvagr ahzore bs fbyhgvbaf juvpu pna or qvivqrq vagb
  gjb pngrtbevrf jurer gur v-gu zrzore pna or erphefviryl qrsvarq va grezf
  bs gur 2 cerivbhf ragevrf.

	pngrtbel 1                              pngrtbel 2
	----------                              ----------

	e  = 1                                  e  = 1
	 1                                       1

	e  = 3                                  e  = 4
	 2                                       2


	Abj gur v-gu (sbe v>2) zrzore bs rnpu pngrtbel pna or sbhaq ol
	gur sbyybjvat erphefvir sbezhyn:

				e    = 4*e  - e
				 v+1      v    v-1

	Gura n gevnatyr bs fvqrf  a, (a+1), (a+2) unf vagrteny nern vs
	naq bayl vs

			2
		a = 12*e  + 1       jurer e  vf n zrzore bs pngrtbel 1
			v                  v

		       2
		a = 6*e  - 3        jurer e  vf n zrzore bs pngrtbel 2
		       v                   v


	Nf fgngrq ol gur bevtvany cbfgre, gur cebbs vf dhvgr vaibyirq
	naq vaibyirf gur fbyhgvba bs fbzr Qvbcunagvar rdhngvbaf.  Vg
	abj nccrnef ubjrire, gung univat gur nobir erphefviryl qrsvarq
	fbyhgvba(f) vg fubhyq or rnfl gb hfr rvgure vaqhpgvba be vasvavgr
	qrfprag gb rfgnoyvfu gur erfhyg.  V unira'g gevrq guvf nccebnpu.