A088848 -id:A088848 - cp's OEIS frontend

A230562 Smallest number that is the sum of 2 positive 4th powers in >= n ways.

0, 2, 635318657

Offset: 0

Jonathan Sondow, Oct 25 2013

Hardy and Wright say that a(3) is unknown.

Guy, 2004: "Euler knew that 635318657 = 133^4 + 134^4 = 59^4 + 158^4, and Leech showed this to be the smallest example. No one knows of three such equal sums."

			0 = (empty sum).
2 = 1^4 + 1^4.
635318657 = 59^4 + 158^4 = 133^4 + 134^4.

R. K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, D1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, 2008; section 21.11.

Carlos Rivera, Puzzle 103: N = a^4 + b^4 = c^4 + d^4

Cf. A003824, A016078, A018786, A046881, A088687, A088848, A088849, A088867, A230561.

A088867 Nonsquarefree elements of A003824, i.e., primitive solutions to a^4 + b^4 = c^4 + d^4 with nonsquarefree value on both sides of the equation.

Original entry on oeis.org

680914892583617, 55683917506335026, 2056314197022256097, 3267700501872475297, 4544031582110882417, 10555434261160919777, 12361929340136667457, 23076050051029379057, 335875812638910622082

Offset: 1

Cino Hilliard, Nov 26 2003

nonn

Original definition was: Numbers that can be expressed as the sum of two distinct 4th powers in exactly two distinct ways that have at least one repeated factor.

Among the first 516 terms of A003824, there are 31 nonsquarefree terms. None of these are expressible in more than 2 ways as sum of two 4th powers. However, some of them, as 4544031582110882417, 12361929340136667457, 335875812638910622082, ..., have gcd(a,b) > 1, for one of the decompositions a^4 + b^4. - M. F. Hasler, Mar 05 2012

			a(1) = A003824(18) = 680914892583617 = 17^2*89*61657*429361 is the first nonsquarefree term of A003824. - _M. F. Hasler_, Mar 05 2012

D. J. Bernstein, List of 516 primitive solutions p^4 + q^4 = r^4 + s^4

Cf. A003824, A088848, A088849.

\ begin a new session and type \r x4data.txt (evaluated Bernstein data) This will allow using %1 as the initial value. omegax4py42(n) = { for (i = 1, n, x = eval( Str("%", i) ); y=omega(x); y1 =bigomega(x); if(y<>y1,print1(x",")) ) }

select(A003824, t->!issquarefree(t))  \\ M. F. Hasler, Mar 05 2012

omega(n)<>bigomega(n) for n = a^4+b^4 = c^4+d^4 for distinct a, b, c, d. n=635318657, 3262811042, ..., 680914892583617, ..., 962608047985759418078417, ...

cp's OEIS Frontend

A230562 Smallest number that is the sum of 2 positive 4th powers in >= n ways.

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