PART 13. Tenth Powers 10.1 Ten terms
10.2 Twelve Terms
10.3 Fourteen Terms
10.4 Sixteen Terms
10.5 Twenty Terms
10.1 Ten Terms No non-trivial solution in the rationals is yet known to equal sums of five 10th powers,
x 10.2 Twelve TermsThe eqn,
x for k = 10 has relatively small solns. The first one was found by Randy Ekl in 1997,
[5, 23, 34, 34, 85, 92] = [16, 25, 28, 32, 71, 95]
There are many more and whether these belong to a parametric family is unknown. However, a family has been found for the even multi-grade system k = 2,4,6,8,10 (pls see update below). A numerical example (not belonging to the known family) was first found in 1999,
{22, 61, 86, 127, 140, 151} = {35, 47, 94, 121, 148, 146}
by Kuosa and Meyrignac as k = 10, and Shuwen (who noticed it was not just for k = 10 but was multi-grade) which automatically leads to an ideal soln of deg 11. To see if this is also for k = 1 if the terms are a-b-c+d-e+f = g-h-i+j-m+n = 13 a+b+c-d-e+f = g+h+i-j-m+n = 53 a+b-c-d+e+f = g+h-i-j+m+n = 161
a+b+c-d+e+f = -g+h-i+j+m+n = 333 a+b-c+d-e+f = -g-h+i+j-m+n = 135 a-b+c+d-e+f = -g-h+i-j+m+n = 185 a-b+c-d+e-f = g-h+i+j-m-n = -91 not counting seven other combinations which simply negate the terms. Also, similar to the known family, it is also the case that, x Whether this is: (1) a fluke, or (2) a property of the possible family which this example belongs to, or (3) solns to even systems with signed terms tend to be valid for k = 1 the more terms there are, is not known. While there are parametric (
x x x
for sextic and octic systems, it is not known if it can be for the decic. (The Choudhry-Wroblewski decic family uses only two.)
Update (June 6, 2009): Choudhry, Wroblewski
It has been brought to my attention (by J. Wroblewski) that the case of an equal sum of
six tenth powers,
a
has already been cracked (in 2008) and shown to have an infinity of solns by reducing it, like the other systems, to a particular elliptic curve, though the Kuusa-Meyrignac-Shuwen soln does not belong to this family. (See their paper " b as, _{i}
a a a a a a
where a variable z has been set z = 1 without loss of generality. Substituting these values into the system, the higher eqns hold if the ff condition is satisfied,
8x
Do the simple change of variables {x, y} = {u, v/(8u
(-17+8u
or a quartic polynomial in
u that is to be made a square. As Choudhry and Wroblewski proved, this is an elliptic curve. One trivial soln is u = 1 but from this initial point, we can compute an infinite number of non-trivial ones such as u = -457/353, etc. However, by looking at the solns a and _{i}b, one can see certain relationships and symmetries between them. Using those relationships, this author will show that it can be simplified to a form analogous to the Letac-Sinha identity for k = 2,4,6,8._{i}(Update, 2/1/10): Choudhry, Wroblewski
The 6th power (k.4.4) identity can be generalized to the form (k.6.6) by adding the italized pair of terms,
[-(axy+bx+cy-d), axy-bx-cy-d, axy-cx+by+d, -(axy+cx-by+d), [-(axy+by+cx-d), axy-by-cx-d, axy-cy+bx+d, -(axy+cy-bx+d),
where again, {x,y} merely swap places. (Since the eqn is homogeneous, one may assume the leading coefficient of the terms as a = 1 without loss of generality.) This obeys the same basic constraints (up to sign changes),
p p (-p p
If we assume,
p p
then it must be the case that 2(b-c) = e-f (eq.1). Choudhry and Wroblewski found for k = 2,4,6,8,10, the equivalent non-trivial solns,
{a,b,c,d,e,f} = {1, 1, 2, 14, 3, 5}, where 2x {a,b,c,d,e,f} = {2, 1, 2, 7, 3, 5}, where 8x
Disregarding eq.1, two other distinct non-trivial solns found (by Piezas), though only up to k = 2,4,6,8, are,
{a,b,c,d,e,f} = {5, 3, 7, 3, 2, 10}, where 125x {a,b,c,d,e,f} = {5, 9, 11, 11, 10, 12}, where 125x
though this author is not certain if these are the same 8th power multi-grades found by Wroblewski. See also Piezas
(a+3b+c) (-a+3b+c)
for k = 2,4,6,8,10, where 45a
where the ratio a/b = {2, 1/2, 3/2, 2/3} must be avoided as it yields trivial solns. Labelling terms as
a and _{i}b, then a small tweak in the sign of {a_{i}_{1}, a_{4}, b_{1}, b_{4}} can make it valid for k = 1 as well. Note that, just like in the Letac-Sinha identity, the two quadratic polynomials to be made squares have palindromic or reversible coefficients. One can easily verify this soln by solving {c,d} in radicals for arbitrary {a,b} and substituting them into the system. Of course, if they are to be rational, then appropriate {a,b} must be chosen, with the smallest non-trivial one being {a,b} = {186, 331} giving,
[886, -293, 1180, 953, 1510, -413] = [700, -107, 1511, 622, 1138, -1075]
after removing a small common factor. By solving the first condition as a quadratic form {a, b} = {u ^{2}+11v^{2}, 2u^{2}+2uv-22v^{2}} using this on the other will result in a quartic polynomial that is to be made a square, with some {u,v} as trivial, but a non-trivial is {u,v} = {12, -5}. Treating this as an elliptic curve, from this initial point, an infinite number of rational solns can then be computed. This identity was found using the relationships between the a and _{i}b and, whether in the Choudhry-Wroblewski version or by this author, they satisfy _{i}seven side conditions: a
_{1}+a_{2} = b_{1}+b_{2} a a a a a
and together with the five eqns,
a
we have 12 (or 11) eqns in 12 unknowns, call this augmented system as
S. By solving this system, it can be shown that the identity given by this author is the _{10+}only non-trivial soln to S. (There may be just 11 eqns as one may be a consequence of the others.) _{10+}Note 1: By negating {a_{1}, a_{4}, b_{1}, b_{4}}, the identity in fact is also valid for k = 1. Also, by eliminating or changing one of the side conditions, it may be possible to come up with a different identity from this one. What would be an appropriate change, I do not yet know.Note 2: I guess my postscript at the final section need to be modified. (However, the case k = 1,3,5,7,9 still needs to be cracked.)(Update, 1/12/10): The identity has the form,
[pa+qb+c, pa+qb-c, qa-pb+d, qa-pb-d, ra+sb, sa-rb]
^{k} =[pa-qb+c, pa-qb-c, qa+pb+d, qa+pb-d, ra-sb, sa+rb]
where ma
{p,q,r,s; m,n} = {1, 3, 2, 8; 45, -11}; for k = 2,4,6,8,10 {p,q,r,s; m,n} = {2, 5, 4, 6; 64/5, -11/5} {p,q,r,s; m,n} = {1, 10, 1, 11; 248/5, -27/5}
See also (Update, 1/11/10)
10.3 Fourteen TermsThe smallest soln to,
x
for k = 10 is,
[1, 8, 15, 26, 26, 33, 38]
given by Ekl (“
[32, -1, -61, -55, -31, -13, -28, 68] [44, -13, -49, -67, 20, -64, 23, 17]
for k = 1,2,4,6,8,10, and belongs to an infinite family. Note that its terms obey the relations,
-32+44 = -1+13 = 61-49 = -55+67 = 12 31+20 = -13+64 = 28+23 = 68-17 = 51
Wroblewski, Piezas
Let a
(8b) (4a+8b)
With terms as {x
x x This was derived from a (k.8.8) discussed in the next section but since two terms (in blue) cancel out, then it reduces to a (k.7.7). However, ratios such as a/b = {4, 12, 12/5, 4/3} should be avoided as the system becomes trivial, with the smallest non-trivial soln as the smallest Pythagorean triple {a,b,c} = {3, 4, 5}, and d = 29. This yields,
[32, -1, -61, -55, -31, -13, -28, 68]
^{k} = [44, -13, -49, -67, 20, -64, 23, 17]^{k} This was cited by Ekl (1998) as the smallest soln in
distinct integers, but it was not mentioned in the paper that it was also multigrade. It is quite interesting that the smallest multigrade (k.7.7) for k = 2,4,6,8,10 belongs to a family, as it is not known if the smallest multigrade (k.6.6) belongs to a family or not.
The two quadratic polynomials generally define an elliptic curve, and there is an infinite number of solns, with other small {a,b} as {612, 35} and {783, 56}. A pair of algebraic forms {a
10.4 Sixteen TermsPiezas, Wroblewski
The latter author suggested the system
[-a-c+x, a+c+x, -a+b+c-y, a-b-c-y, -a-b-c-t, a+b+c-t, -b+c+z, b-c+z] [a-c+x, -a+c+x, a+b+c-y, -a-b-c-y, -a+b-c-t, a-b+c-t, b+c+z, -b-c+z]
which, for some {a,b,c,x,y,z,t}, is valid for k = 1,2,4,6,8,10. This has a
x x x x
An example by Wroblewski is,
[-31, 193, 75, -100, -179, 164, 72, 51]
which is for k = 1,2,3,4,6,8,10. The former author found the complete radical soln using
p-s = (-2b+9b q-s = 8(1-b)(1-c)(-b+c)c/(3b+3c+3bc) q-r = (-9+2b-2c-8bc+8c
and it is easy to solve for {p,q,r}. Substituting these into the system and factoring at either k = 6,8,10, one will find a common
{a,b,c} = {1, (u+12v)/(4u), (u-12v)/(4u)}
then {p,s} are squares, {q,r} become just quadratic polynomials, and two terms {x (Update, 1/17/10)
10.5 Twenty TermsWroblewski added four more terms to each side of the (k.6.6) to get the (k.10.10),
[pa+qb+c, pa+qb-c, qa-pb+d, qa-pb-d, ra+sb, sa-rb,
ta+ub, ua-tb, va+wb, wa-vb]^{k} =[pa-qb+c, pa-qb-c, qa+pb+d, qa+pb-d, ra-sb, sa+rb,
for k = 2,4,6,8,10, where ma
{p,q,r,s,t,u,v,w; m,n} = {4, 4, 3, 11, 5, 7, 7, -3; 45, -11} {p,q,r,s,t,u,v,w; m,n} = {4, 4, 9, 17, 15, 13, 13, 7; 189, 85} {p,q,r,s,t,u,v,w; m,n} = {3, 3, 11, 17, 16, 14, 14, 10; 220, 136}
though certain ratios
(Update, 3/1/10): Piezas
This author noticed that, other than the obvious relations p = q, u = v, the above also obey,
r = -q+v s = q+v mn = rstw
Combined with the conditions k = 2,4,6,8,10, these were enough to find a formula for these variables. Thus,
{p,q,r,s; t,u,v,w} = {y, y, -y+z, y+z; x+y, z, z, x-y} {m,n} = {x
where {x,y,z} satisfies the simple quadratic condition
{x,y} = {1, 4}; {11, 4}; {13, 3}
However, there is an infinite number of them given by {x,y,z} = {e
{p,q,r,s; t,u,v,w; m,n} = {5, 5, 9, 19; 16, 14, 14, 6; 216, 76}
with one soln to 216a
{a,b} = {495753715, 352750681}
though presumably smaller solns may exist. From this initial rational point, one can then compute an infinite more. And so on for an infinite number of other formulas using different {m,n} and other elliptic curves, though expanding this family at k = 12 or 14 generates only trivial solns. ( |