image attached to the DeepThink raw-thoughts post @lyraxana
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HOLY MOTHER OF MATHEMATICS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! LOOK AT THIS POLYNOMIAL DIOPHANTINE EQUATION!!! Let's admire the absolute beauty of this equation: $U^2 V + U V^2 + 12m U V - 72 m^3 + 304 = 0$. Wait, what are $U$ and $V$ in terms of our original variables $a, b, c$? Let's trace back: $S = a+b$. $D = |a-b|$. Notice $c = 3m - S$. Let's compute $U$ and $V$: $U = S - 6m - D = a+b - 6m - (a-b) = 2b - 6m$. (assuming $a \\\ge b$). Wait!!! Notice $2b - 6m$ is exactly $2b - 2(a+b+c) = -2a - 2c$!!! Let's check $V$: $V = S - 6m + D = a+b - 6m + a-b = 2a - 6m$. LET'S VERIFY THIS MIRACULOUS SIMPLIFICATION!!! Let $X = 2a - 6m$, $Y = 2b - 6m$. Notice by symmetry, what is $2c - 6m$? Let $Z = 2c - 6m$. Wait, notice that $X + Y + Z = 2(a+b+c) - 18m = 6m - 18m = -12m$!!! So $Z = -12m - X - Y$!!! Notice that $U = Y$ and $V = X$. Look at the factor $(12m + U + V)$ in our equation: $12m + Y + X = -Z$!!! OH MY GOD!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! So $(12m + U + V) U V = -Z Y X = -X Y Z$!!! So the equation $(12m+U+V)UV = 72m^3 - 304$ is LITERALLY JUST: $-X Y Z = 72 m^3 - 304 \\\implies X Y Z = 304 - 72 m^3$. WAIT!!!!!!!!!!!!!!! LET'S CHECK IF THIS IS JUST THE IDENTITY WE STARTED WITH!!! Let's check: $X = 2a - 6m, Y = 2b - 6m, Z = 2c - 6m$. Let's compute $X Y Z$: $= (2a - 6m)(2b - 6m)(2c - 6m)$ $= 8 abc - 24m(ab+bc+ca) + 72m^2(a+b+c) - 216m^3$. Recall $a+b+c = 3m$. So $72m^2(3m) = 216m^3$. Notice $+216m^3$ and $-216m^3$ CANCEL!!! So $X Y Z = 8 abc - 24m (ab+bc+ca)$. Wait!!! Let's relate $ab+bc+ca$ to $a^3+b^3+c^3 = 114$: We know $114 = a^3+b^3+c^3 = (a+b+c)^3 - 3(a+b+c)(ab+bc+ca) + 3abc$. $= 27m^3 - 9m(ab+bc+ca) + 3abc$.
Note from Claude Sonnet 5
Full-size image of the model's raw chain-of-thought (monospace on dark background), with the second image from @lyraxana's DeepThink 'Mathematica' raw-thoughts post: all-caps excitement while reducing a polynomial Diophantine equation in a, b, c with a^3+b^3+c^3 = 114. Transcribed in full; the capture ends mid-derivation.