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Least positive integer x such that x and n*x are both differences of fourth powers.
1

%I #19 Sep 04 2013 23:02:58

%S 1,179727600,80,1040,16,2320,4080,236187120,76960,240,17680,76960,80,

%T 1040,1,1,15,65520,4851120,224991600,100880,1728480,27120,1389920,

%U 19578624,1048560,240,2986560,80,80,2465,11232975,65,16,80,2320,12240,707200,16,6560

%N Least positive integer x such that x and n*x are both differences of fourth powers.

%C It's not obvious that a(n) exists for all n.

%C a(967) > 8*10^15 (if it exists). - _Donovan Johnson_, Sep 04 2013

%D A. Choudhry, Indian J. pure appl. Math. 26(11) (1995), 1057-1061

%H Robert Israel and Donovan Johnson, <a href="/A228760/b228760.txt">Table of n, a(n) for n = 1..966</a> (first 205 terms from Robert Israel)

%H Tito Piezas II, <a href="http://math.stackexchange.com/questions/482523/is-the-quartic-diophantine-equation-a4nb4-c4nd4-solvable-for-any-integ">Is the quartic diophantine equation a^4+n*b^4 = c^4+n*d^4 solvable for any integer n?</a>

%e For n = 3, 80 = 3^4 - 1^4 and 3*80 = 4^4 - 2^4.

%p T:= 10^12; N:= 100; # to get solutions with n*a(n)<=T and n <= N

%p cmax := floor(fsolve('c'^4 - ('c'-1)^4 = T));

%p S:= {seq(seq(c^4 - a^4, a = ceil((max(0,c^4 - T))^(1/4))..c-1),c=1..cmax)}:

%p for n from 1 to N do

%p B:= S intersect map(`*`,S,n);

%p if B <> {} then

%p A[n]:= min(B)/n;

%p printf("a[%d] = %d\n",n,A[n]);

%p end if

%p end do: # _Robert Israel_, Sep 02 2013

%Y Cf. A152044.

%K nonn

%O 1,2

%A _Robert Israel_, Sep 02 2013