A036236 and A078457
Hans Havermann
pxp at rogers.com
Thu Jul 17 19:34:59 CEST 2008
still there: Entropy!
says nothing about existence, since a(n) = infinity is implied
shorthand for non-existence.
street) here:
Return-Path: <jvospost3 at gmail.com>
X-Ids: 164
DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed;
d=gmail.com; s=gamma;
h=domainkey-signature:received:received:message-id:date:from:to
:subject:mime-version:content-type:content-transfer-encoding
:content-disposition;
bh=e06IyL0Vv8bK2nMsGPaNM9hhC0dXw67rIRNwZ9GBwx8=;
b=JNe/JVZkS7vcesEds/Q2fOR7f2bSmG22FVa8O/WX861COKPIknUkp3YMdmg+x0fc35
0BHE/1C1juvWsB5KnlEw+kBzgo7jtJY9CsZMsd81VDWd3RMy4B+9QwEl3fNWBbGJw3NM
l7hn19oysG59fRpx0O3B4nwm4Ad4y6fObFSL0=
DomainKey-Signature: a=rsa-sha1; c=nofws;
d=gmail.com; s=gamma;
h=message-id:date:from:to:subject:mime-version:content-type
:content-transfer-encoding:content-disposition;
b=kA0WK+RWwLMQXGoB+jf+9U+XV4C6/aGpuqdC/W1+m1t3MZV8rL4y6dmPLvJUwWNBnu
zsa8ahWZvxtcPzfCj1Mzetik0jRgE4mHtgFJEvVbE1glZu9+70ZohtEtLY8wJxecPMwr
HXN91RH+VY0IeIIk13VmcrCisj1SwSCe/0O2E=
Message-ID: <5542af940807171156p7b946036y5e4750d6773e94da at mail.gmail.com>
Date: Thu, 17 Jul 2008 11:56:19 -0700
From: "Jonathan Post" <jvospost3 at gmail.com>
To: SeqFan <seqfan at ext.jussieu.fr>
Subject: Sequence suggested: sum of at most 4 nonzero 4-th powers in more than one way
MIME-Version: 1.0
Content-Type: text/plain; charset=ISO-8859-1
Content-Transfer-Encoding: 7bit
Content-Disposition: inline
X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-4.0 (shiva.jussieu.fr [134.157.0.164]); Thu, 17 Jul 2008 20:56:22 +0200 (CEST)
X-Virus-Scanned: ClamAV 0.93/7417/Tue Jun 10 03:14:29 2008 on shiva.jussieu.fr
X-Virus-Status: Clean
X-Miltered: at jchkmail2.jussieu.fr with ID 487F79B6.006 by Joe's j-chkmail (http : // j-chkmail dot ensmp dot fr)!
X-j-chkmail-Enveloppe: 487F79B6.006/66.249.82.237/wx-out-0506.google.com/wx-out-0506.google.com/<jvospost3 at gmail.com>
X-j-chkmail-Score: MSGID : 487F79B6.006 on jchkmail2.jussieu.fr : j-chkmail score : X : R=. U=. O=# B=0.222 -> S=0.260
X-j-chkmail-Status: Unsure
I'm looking to make a sequence derived as a special subset of A004833
Numbers that are the sum of at most 4 nonzero 4-th powers.
This would be "numbers that are the sum of (exactly) or (no more than)
4 nonzero 4-th powers in more than one way."
This can happen in 4 ways, as exemplified in
Piezas, Tito III and Weisstein, Eric W. "Diophantine Equation--4th
Powers." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/DiophantineEquation4thPowers.html
(i) A^4 + B^4 + C^4 + D^4 = E^4 (the 4.1.4 equation).
Smallest example is 30^4 + 120^4 + 272^4 + 315^4 = 353^4 = 15527402881.
(ii) A^4 + B^4 + C^4 + D^4 = E^4 + F^4 (the 4.2.4 equation).
Parametric solutions are known. Smallest example is
5^4 + 5^4 + 6^4 + 8^4 = 3^4 + 9^4 = 6642 (Ramanujan)
If we drop the "nonzero" we have
3^4 + 5^4 + 8^4 + 0^4 = 7^4 + 7^4 = 4802 (Ramanujan)
(iii) A^4 + B^4 + C^4 + D^4 = E^4 + F^4 + G^4 (the 4.3.4 equation).
Parametric solutions are known. Smallest example is
4^4 + 4^4 + 5^4 + 6^4 = 2^4 + 2^4 + 7^4 = 2433 (Ramanujan).
Ramanujan also gave:
7^4 + 8^4 + 10^4 + 13^4 = 3^4 + 9^4 + 14^4 = 45058.
5^4 + 5^4 + 6^4 + 14^4 = 7^4 + 10^4 + 13^4 = 40962.
(iv) A^4 + B^4 + C^4 + D^4 = E^4 + F^4 + G^4 + H^4 (the 4.4.4 equation).
I don't know the smallest.
If we drop the "exactly" for the "at most" we have values
corresponding to solutions
A^4 + B^4 + C^4 = D^4 (4.1.3 equation)
Cf. (N. Elkies, 1987) and (Roger Frye, 1988) originally thought
nonexistent (also known as the Euler quartic conjecture).
and solutions of A^4 + B^4 = C^4 + D^4 (4.2.2 equation)
Cf. A003824, A018786.
A003824 Sum of two 4th powers in more than one way (primitive solutions).
635318657, 3262811042, 8657437697, 68899596497, ...
A018786 Sum of two 4th powers in more than one way.
635318657, 3262811042, 8657437697, 10165098512, ...
So is there an interesting sequence which begins:
2433, 4802, 6642?
If so, what is a(4)?
If the sequence A139334 is continued following the rules in the definition, we get:
2, 24, 28, 311, 312, 325, 337, 340, 365, 398, 405, 439,...
with the first differences A139334:
The problem that arises is: the next difference, a139334(12) requires a leading 0
to match the final 0 in a(8)=340; this cannot be included in the OEIS.
I have no access to the Angelini article and cannot say how it works around that
case.
Richard
More information about the SeqFan
mailing list