Prime as sum of three numbers whose product is a cube

To add pessimism in finding such prime $p$ that cannot be written as sum of $3$ co-divisors of cube, I've drawn images:

Number of ways to write an integer number $p$ as the sum $x+y+z$, where $x\cdot y\cdot z=c^3$.

($\color{red}{\bf{red}}$ dots $-$ composite numbers, $\bf{black}$ dots $-$ prime numbers)

$$p\le 500$$

$$p\le 5\;000$$

$$p\le 50\;000$$

$$p\le 500\;000$$

$$p\le 5\;000\;000$$

Control points:
$p=486$ (composite): $1$ way: $486 = 162+162+162$;
$p=2048$ (composite): $2$ ways: $2048 = 128+720+1200 = 224+256+1568$;
$p=6656$ (composite): $3$ ways: $\small {6656 = 416+2340+3900 = 512+1536+4608 = 728+832+5096}$;
$p=7559$ (prime): $4$ ways: $\scriptsize{7559 = 9+50+7500 = 114+225+7220 = 135+1024+6400 = 722+2809+4028}$;
$p=26624$ (composite): $5$ ways;
$p=58757$ (prime): $10$ ways;
$p=80429$ (prime): $13$ ways;
$p=111611$ (prime): $15$ ways;
$...$

These images were made in style like here.
(Since I've seen images for Goldbach's conjecture, I lost any hope to find contradiction for it).

Method of search:

to build array $a[3N]$ for storing number of ways;

$a[p]$ is the number of ways to write $p$ as such sum; (initially, each $a[j]=0$);

for $c = 1 ... N$
$\quad$ create list of prime divisors of $c^3$;
$\quad$ (there is enough to know prime decomposition of $c$)
$\quad$ for each pair $x,y$ of divisors of $c^3$ to find $z=\dfrac{c^3}{xy}$;
$\quad$ if $z\in\mathbb{Z}$ and if $x\le y\le z$, then increase $a[x+y+z]$.

(if $c>N$, then sum of co-divisors of $c^3$ is greater than $3N$).

Two comments on positivity in David Speyer's method.

First, we can take $x,y > 0$ in $p = x^2 + x y + y^2,$ which we can do when $p \equiv 1 \pmod 3.$ Given $p = u^2 + u v + v^2,$ with $u>0$ but $v < 0.$ If $u+v > 0,$ we can use $p = (u+v)^2 + (u+v)(-v) + (-v)^2$ to get everything positive, as both $u+v, -v > 0.$ If $u+v <0,$ switch to $u^2 + u(-u-v)+ (-u-v)^2.$

Next, we get another good one with an indefinite form $$\color{blue}{x^2 + 4 x y + 2 y^2.}$$

Lemma: every (positive) prime $p \equiv \pm 1 \pmod 8 $ can be written as $p = u^2 - 2 v^2$ with $u > 2v.$

Proof. Take the representation $p = u^2 - 2 v^2$ such that $v$ has the smallest possible positive value, with $u>0.$ Note that $u > v \sqrt 2,$ also $u$ is odd. We get a slightly different representation with $(u,-v).$ Next, we apply the generator of the automorphism group of $x^2 - 2 y^2$ to get another representation, $$ (3u-4v, 2u-3v). $$

Now, if $2u - 3v \leq 0, $ by minimality of $|v|$ we also get $2u - 3 v \leq -v,$ so $2u \leq 2v$ and $u \leq v.$ This contradicts $u > v \sqrt 2.$

Therefore, $2u - 3 v > 0,$ and minimality again says $2u-3v > v.$ This gives us $2u>4v$ and $u > 2v.$

So we have $u > 2 v.$ Define $$ x = u - 2 v, \; \; y = v. $$ Then $$ p = x^2 + 4 x y + 2 y^2 = u^2 - 2 v^2, $$ with both $x,y > 0.$

EDIT, Thursday, August 21. The remaining primes are $5,11 \pmod 24.$ These are all represented by $2x^2 + 3 y^2,$ but the misfortune is that this does not give any cubes; furthermore, it appears that no $SL_2 \mathbb Z$-equivalent form has a product of the three coefficients a cube. So, the best i have come up with is $$ 14 x^2 + 36 xy + 147 y^2. $$ The good part is the cubes. there are two bad parts: it does represents only primes $5,11 \pmod {24},$ but the additional constraint is that the prime not be a quadratic residue $\bmod {17}.$ Furthermore, with a positivity constraint, as in the original problem, it represents only about half of those. So, all told, it gives only about one fourth of the remaining primes, such as $$ 197, 677, 1907, 1979, 2213 , 2237, 3083, 3803, 4091, 6011, 7349, 8429 , 10139 , 10781, 11213, \ldots $$

So, not bad but not great, about 13/16 of the primes so far.

EDIT, Friday, August 22.

It appears that all primes $p > 0$ with $(p|5) = 1$ and $(p|11) = 1$ are represented by $$ x^2 + 25 xy + 5 y^2 $$ with $x,y > 0.$ I think I have the proof; it starts with an elementary argument that the same can be accomplished for $x^2 + 23 xy- 19 y^2,$ then some fiddling with improper automorphs similar to the proof for $x^2 + 4 x y + 2 y^2$ but with more steps.

EEEDDDIIIITTTTTTT, Saturday, August 23:

THEOREM: given a prime $p > 30$ with $(p|5)=1$ and $(p|11)=1,$ we can write $ p = x^2 + 25 x y + 5 y^2 $ with positive integers $x,y.$

Take prime $p > 0$ and integers $s,t > 0$ with $$ p = s^2 - 605 t^2. $$ Note that $s > t \sqrt {605} \approx 24.5967 t.$Then we can write $$ p = x^2 + 23 x y - 19 y^2, $$ with $x = s - 23 t, \; \; y = 2 t,$ so that $x,y >0.$

Now, find the smallest positive integer $v$ such that there is another positive integer $u$ with $$ p = u^2 + 23 u v - 19 v^2. $$

Lemma: given $ p = x^2 + 23 x y - 19 y^2 $ with $y < 0,$ then $y \leq -v.$

Proof: if $x < 0,$ we get a representation in positive integers $(-x,-y),$ so $-y \geq v.$ If $x > 0,$ note that $$ x > \frac{\sqrt {605} + 23}{2} |y| \approx 23.798 |y|. $$ Therefore the new representation $(x + 23 y, -y)$ is again in positives and $-y \geq v.$

Corollary: if $ p = x^2 + 23 x y - 19 y^2 $ with $x <0, y >0,$ then $y \geq v.$

Back to the minimum $(u,v),$ both positive. Note $$ u > \frac{\sqrt {605} - 23}{2} v \approx 0.798 v. $$

There is another representation, $$(4u-3v,5u-4v)$$ We know $5u-4v \neq 0$ as $p$ is not a square. If $5u-4v < 0,$ then $5u-4v \leq -v,$ thus $5u < 3 v.$ But this gives $u < 0.6 v,$ while in fact, $u > 0.798 v. $

Therefore, $5u-4v > 0,$ whereupon $5u - 4 v \geq v.$ Thus $5 u \geq 5 v$ and $u \geq v.$ Finally, as $u,v$ are relatively prime, they are not equal unless both are actually $1,$ giving the prime $5.$ Thus, for $p > 5,$ we get $u > v.$

Finally, finally, finally, we can write $$ \color{blue}{ p = x^2 + 25 x y + 5 y^2} $$ with $$ x = u - v, y = v $$ both positive.

I wrote a small script in MATLAB and verified your claim for all the primes less than $150,000$. Only $2,5,11$ do not satisfy your claim. For many of the primes, there are multiple ways to write it as a sum of three number such that its product is a cube. My script just looks for the first occurrence of such triplets for each prime.

Here is a .txt file with the "first" set of triplets for primes less than $150,000$.

Here is an attempt that doesn't quite work. In this post, $\left( \frac{a}{p} \right)$ is the quadratic residue symbol.

If $\left( \frac{-3}{p} \right) = 1$, then $p$ is of the form $x^2+xy+y^2$.

If $\left( \frac{85}{p} \right) = 1$, then $p$ is either of the form $9 x^2 + 25 xy + 15 y^2$ or $3 x^2 + 25 xy + 45 y^2$.

If $\left( \frac{-255}{p} \right) = 1$, then $p$ is of one of the forms $x^2+xy+64 y^2$, $2 x^2 + xy + 32 y^2$, $4 x^2 + xy + 16 y^2$, $8 x^2 + xy + 8 y^2$, $3 x^2 + 3 xy + 22 y^2$, $5 x^2 + 5 xy + 14 y^2$, $6 x^2 + 3 xy + 11 y^2$ or $7 x^2 + 5 xy + 11 y^2$.

Now, one of the three quadratic residues must be $1$, since $\left( \frac{-255}{p} \right) = \left( \frac{-3}{p} \right)\left( \frac{85}{p} \right)$. And for a lot of the quadratic forms above, we win: $(x^2)(xy)(y^2)$, $(9 x^2)(25 xy)(15 y^2)$, $(3 x^2)(25 xy)(45 y^2)$, $(x^2)(xy)(64 y^2)$, $(2 x^2)(xy)(32 y^2)$, $(4 x^2)(xy)(16 y^2)$ and $(8 x^2)(xy)(8 y^2)$ are all cubes. Unfortunately, the last $4$ quadratic forms of discriminant $-255$ don't work when written in reduced form. I haven't yet done a serious search to see whether some change of basis might put them into the form $a x^2 + bxy + c y^2$, with $abc$ a cube.

Also, we don't know what the signs of $x$ and $y$ are, so this doesn't necessarily get us three positive integers whose product is a cube.

Still, I am optimistic that we might be able to find enough quadratic forms of the form $a x^2 + bxy + c y^2$, with $abc=1$, to cover all the primes, at least if we ignore the sign issue.

I wrote the following JavaScript which is designed to run in Windows Script host:

• isprime(n) - determine whether a number is prime via brute force method
• showmagic(f, txt) - writes a line to both screen and text file
• findmagic(f, n) - detects and writes the magic for a given prime
• main - tests the first 500 numbers

Save the script to a JavaScript file, say magic.js and invoke cscript magic.js from the command prompt to execute the script using the console version of the Windows Script Host. Because it's written in JavaScript, the script gets exponentially slower to run if you set it to test the first 10000 numbers.

function isprime(n) {
  if (n < 2) return false;
  for (var i=2; i*i<=n; i++)
    if (n % i === 0) return false;
  return true;
}

function showmagic(f, txt) {
  f.WriteLine(txt);
  WScript.Echo(txt);
}

function findmagic(f, n) {
  for (var i=1; i<=n-2; i++) {
    for (var j=1; j<=n-1-i; j++) {
      var k=n-i-j;
      var c3=i*j*k;
      for (var c=1; c*c*c<=c3; c++) {
        if (c*c*c === c3) {
          showmagic(f, n + "=" + i + "+" + j + "+" + k + " ; " + i + "*" + j + "*" + k + "=" + c + "^3");
          return;
        }
      }
    }
  }
  showmagic(f, n + "=no magic");
}

var fso = new ActiveXObject("Scripting.FileSystemObject");
var f = fso.CreateTextFile("magic.txt", true);
for (var n=1; n<=500; n++)
  if (isprime(n))
    findmagic(f, n);
f.Close();