TCD_talk_9th_Feb_200512.html

Cube-square identities

> restart;

Odd squares first

> # see ?polynomial for Maple's use of 'quo' and 'rem' with polynomials

> expand(((27*x^2 + 27*x + 7 - 1)/3)^3);

(11.1)

Now, this was a bit odd, to say the least:

> quo(729*x^6+2187*x^5+2673*x^4+1701*x^3+594*x^2+108*x+8,
27*x^2+27*x+7, x, 'R0');

(11.2)

> R0;

(11.3)

> floor(31/27);

(11.4)

I had to resort to this to get the odd square identity:

> (729*x^6+2187*x^5+2673*x^4+1701*x^3+594*x^2+108*x+8)
- expand((27*x^2+27*x+7)*(27*x^4+54*x^3+38*x^2+11*x+1));

(11.5)

> factor(4*x^2+4*x+1);

(11.6)

A look at the quotient quartic:

> factor(27*x^4+54*x^3+38*x^2+11*x+1);

(11.7)

Thus one has the following identity:

Now the even squares (seven cases)

(0) The zero-th of the seven even-square producing quadratics:

> expand(((108*x^2 + 1 - 1)/3)^3);

(11.8)

> quo(46656*x^6, 108*x^2+1, x, 'r0');

(11.9)

> r0;

(11.10)

I had to resort to this to get the zero-th of the even square identities:

> (46656*x^6) - expand((108*x^2+1)*(432*x^4-4*x^2));

(11.11)

That, of course, is identically .

Thus one has the following identity:

(1) The i = 1 case of the seven even-square producing quadratics:

> expand(((756*x^2+648*x+139 - 1)/3)^3);

(11.12)

> quo(16003008*x^6+41150592*x^5+44035488*x^4+25100928*x^3+8038224*x^2
+1371168*x+97336, 756*x^2+648*x+139, x, 'r1');

(11.13)

> 16003008*x^6+41150592*x^5+44035488*x^4+25100928*x^3
+8038224*x^2+1371168*x+97336 - expand((756*x^2+648*x+139)*
(21168*x^4+36288*x^3+23252*x^2+6600*x+700));

(11.14)

where the constant term '700' is obtained from the 'r1' quotient constant term:

> floor(18907/27);

(11.15)

> factor(196*x^2+168*x+36);

(11.16)

is, in fact, . Thus one has the following identity:

A look at the quotient quartic:

> factor(21168*x^4+36288*x^3+23252*x^2+6600*x+700);

(11.17)

That quadratic has discriminant:

> 162^2 - 4*189*35;

(11.18)

is . Interesting...

(2) The i = 2 case of the seven even-square producing quadratics:

> expand(((756*x^2+864*x+247 - 1)/3)^3);

(11.19)

> quo(16003008*x^6+54867456*x^5+78327648*x^4+59595264*x^3+25487568*x^2+
5809536*x+551368, 756*x^2+864*x+247, x, 'r2');

(11.20)

> 16003008*x^6+54867456*x^5+78327648*x^4+59595264*x^3+25487568*x^2+
5809536*x+551368 - expand((756*x^2+864*x+247)*
(21168*x^4+48384*x^3+41396*x^2+15712*x+2232));

(11.21)

where the constant term '2232' is obtained from the 'r2' quotient constant term:

> floor(60271/27);

(11.22)

> factor(196*x^2+224*x+64);

(11.23)

is, in fact, ... . Thus one has the following identity:

A look at the quotient quartic:

> factor(21168*x^4+48384*x^3+41396*x^2+15712*x+2232);

(11.24)

That quadratic has discriminant:

> 216^2 - 4*189*62;

(11.25)

is also ...

(3) The i = 3 case of the seven even-square producing quadratics:

> expand(((1404*x^2+864*x+133 - 1)/3)^3);

(11.26)

> quo(102503232*x^6+189236736*x^5+145364544*x^4+59470848*x^3+13666752*x^2
+1672704*x+85184, 1404*x^2+864*x+133, x, 'r3');

(11.27)

> 102503232*x^6+189236736*x^5+145364544*x^4+59470848*x^3+13666752*x^2
+1672704*x+85184 - expand((1404*x^2+864*x+133)*
(73008*x^4+89856*x^3+41324*x^2+8416*x+640));

(11.28)

where the constant term '640' is obtained from the 'r3' quotient constant term

> floor(17293/27);

(11.29)

> factor(676*x^2+416*x+64);

(11.30)

is, in fact, ... . Thus one has the following identity:

A look at the quotient quartic:

> factor(73008*x^4+89856*x^3+41324*x^2+8416*x+640);

(11.31)

Ah! Interesting! Two quadratics. That's novel...

The first dfiscriminant:

> 216^2 - 4*351*32;

(11.32)

Oh! That's ...

And the other one:

> 32^2 - 4*52*5;

(11.33)

Oh! That's ...

(4) The i = 4 case of the seven even-square producing quadratics:

> expand(((1404*x^2+1944*x+673 - 1)/3)^3);

(11.34)

> quo(102503232*x^6+425782656*x^5+736729344*x^4+679684608*x^3+352622592*x^2
+97542144*x+11239424, 1404*x^2+1944*x+673, x, 'r4');

(11.35)

> 102503232*x^6+425782656*x^5+736729344*x^4+679684608*x^3+352622592*x^2
+97542144*x+11239424 - expand((1404*x^2+1944*x+673)*
(73008*x^4+202176*x^3+209804*x^2+96696*x+16700));

(11.36)

where the constant term '2232' is obtained from the 'r4' quotient constant term:

> floor(450913/27);

(11.37)

> factor(676*x^2+936*x+324);

(11.38)

is, in fact, ... . Thus one has the following identity:

A look at the quotient quartic:

> factor(73008*x^4+202176*x^3+209804*x^2+96696*x+16700);

(11.39)

Again, two more quadratics...

The first discriminant:

> 486^2 - 4*351*167;

(11.40)

is again ,

while the other one is:

> 72^2 - 4*52*25;

(11.41)

is once again ...

(5) The i = 5 case of the seven even-square producing quadratics:

> expand(((2052*x^2+864*x+91 - 1)/3)^3);

(11.42)

> quo(320013504*x^6+404227584*x^5+212308128*x^4+59346432*x^3+9311760*x^2
+777600*x+27000, 2052*x^2+864*x+91, x, 'r5');

(11.43)

> 320013504*x^6+404227584*x^5+212308128*x^4+59346432*x^3+9311760*x^2
+777600*x+27000 - expand((2052*x^2+864*x+91)*
(155952*x^4+131328*x^3+41252*x^2+5728*x+296));

(11.44)

where the constant term '2232' is obtained from the 'r4' quotient constant term:

> floor(8011/27);

(11.45)

> factor(1444*x^2+608*x+64);

(11.46)

is, in fact, ... . Thus one has the following identity:

A look at the quotient quartic:

> factor(155952*x^4+131328*x^3+41252*x^2+5728*x+296);

(11.47)

Ah! It's irreducible...

(6) The i = 6 case of the seven even-square producing quadratics:

> expand(((2052*x^2+3240*x+1279 - 1)/3)^3);

(11.48)

> quo(320013504*x^6+1515853440*x^5+2991372768*x^4+3147880320*x^3
+1863047952*x^2+587982240*x+77308776, 2052*x^2+3240*x+1279, x, 'r6');

(11.49)

> 320013504*x^6+1515853440*x^5+2991372768*x^4+3147880320*x^3
+1863047952*x^2+587982240*x+77308776 - expand((2052*x^2+3240*x+1279)*
(155952*x^4+492480*x^3+582980*x^2+306600*x+60444));

(11.50)