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If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides by Noam Elkies where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer with bringing this problem under his attention.)

It seems that the credit for this goes to Euler. The surface described by the equation above has a K3 surface with maximal Picard rank ($\rho=20$) as its smooth model. Euler presumably used the elliptic fibrations and the presence of many rational curves on the surface to produce a $1$-parameter family of solutionsolutions (as suggested by Will Sawin).

If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides by Noam Elkies where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer with bringing this problem under his attention.)

It seems that the credit for this goes to Euler. The surface described by the equation above has a K3 surface with maximal Picard rank ($\rho=20$) as its smooth model. Euler presumably used the elliptic fibrations and the presence of many rational curves on the surface to produce a $1$-parameter family of solution (as suggested by Will Sawin).

If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides by Noam Elkies where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer with bringing this problem under his attention.)

It seems that the credit for this goes to Euler. The surface described by the equation above has a K3 surface with maximal Picard rank ($\rho=20$) as its smooth model. Euler presumably used the elliptic fibrations and the presence of many rational curves on the surface to produce a $1$-parameter family of solutions (as suggested by Will Sawin).

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R.P.
R.P.
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It seems that Euler has completely answered your first question: seeIf for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides to a talk given by Noam Elkies, who where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer (presumably forwith bringing this problem under his attention).)

It seems that the credit for this goes to Euler. The surface described by Chris Wuthrichthe equation above has a K3 surface with maximal Picard rank ($\rho=20$) as its smooth model. Euler usespresumably used the elliptic fibrations and the presence of many rational curves on itthe surface to produce infinitely many solutionsa $1$-parameter family of solution (as suggested by Will Sawin).

[In the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, the equation becomes the more symmetric $$ pqr(p+q+r) = A^2. $$ I knew I recognized this equation from somewhere, and after a little searching I hit upon Noam's presentation.]

It seems that Euler has completely answered your first question: see the slides to a talk given by Noam Elkies, who credits Franz Lemmermeyer (presumably for bringing this under his attention).

The surface described by Chris Wuthrich has a K3 surface as its smooth model. Euler uses the presence of rational curves on it to produce infinitely many solutions (as suggested by Will Sawin).

[In the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, the equation becomes the more symmetric $$ pqr(p+q+r) = A^2. $$ I knew I recognized this equation from somewhere, and after a little searching I hit upon Noam's presentation.]

If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$ pqr(p+q+r) = A^2. $$ Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides by Noam Elkies where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer with bringing this problem under his attention.)

It seems that the credit for this goes to Euler. The surface described by the equation above has a K3 surface with maximal Picard rank ($\rho=20$) as its smooth model. Euler presumably used the elliptic fibrations and the presence of many rational curves on the surface to produce a $1$-parameter family of solution (as suggested by Will Sawin).

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R.P.
R.P.
  • 4.7k
  • 19
  • 43
  • 67

It seems that Euler has completely answered your first question: see the slides to a talk given by Noam Elkies, who credits Franz Lemmermeyer (presumably for bringing this under his attention).

The surface described by Chris Wuthrich has a K3 surface as its smooth model. Euler uses the presence of rational curves on it to produce infinitely many solutions (as suggested by Will Sawin).

[In the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, the equation becomes the more symmetric $$ pqr(p+q+r) = A^2. $$ I knew I recognized this equation from somewhere, and after a little searching I hit upon Noam's presentation.]