Question

Let’s define 26 to be a sandwich number because it is sandwiched between a perfect cube and a perfect square. That is, 26−1 = 25 = 5^2 and 26 + 1 = 27 = 3^3. Are there any other sandwich numbers? That is, can we find all integer solutions to the Diophantine equation y^3 − x^2 = 2? (I suppose my definition of sandwich numbers didn’t specify that the cube was the larger number and the square was smaller, so maybe we should also try to find solutions to y^3 − x^2 = −2 – but to keep this problem a little shorter, let’s just look at y^3 − x^2 = 2.) This problem answers the question with the use of quadratic integers! (a) First, show that if y^3 − x^2 = 2, then y must be odd (consider the equation mod 8). From this, show that x must also be odd. (b) Next, rewrite the equation as y^3 = x^2 + 2. The right-hand side can be factored in the quadratic field Q[√−2] as (x + √−2)(x − √−2). By Theorem 8.20 in the book, we know this is a Euclidean domain and thus a UFD. Use Theorem 8.5 (at the end of the question) to characterize the quadratic integers in Q[−2]. Also, list all the units in Q[√−2]. Is 2 prime in Q[√−2]? (c) If x is any odd standard (rational) integer, what can be said about the greatest common factor of the quadratic integers x + √−2 and x − √−2 in Q[√−2]? (d) If α and β are relatively prime quadratic integers in Q[√−2] and their product is a perfect cube, must α and β themselves be perfect cubes? (Note that you have already found all possible units in Q[√−2]). (e) If x is any odd standard (rational) integer and a and b are standard (rational) integers for which x + √−2 = (a + b√−2)^3, use the binomial expansion of the right-hand side to solve for b, then a, and ultimately, x. List all possible solutions in standard (rational) integers for y^3 − x^2 = 2. Theorem 8.5: If d is not congruent to 1 (mod 4), then the integers of Q[sqrt(d)] are exactly those numbers of the form a + b√(d), where a and b are rational integers. If d is congruent to 1 (mod 4), then the integers of Q[sqrt(d)] are those numbers of the form (a + b√(d))/2, where a and b are rational integers, both even or odd.

          Let’s define 26 to be a sandwich number because it is sandwiched between a perfect cube and a perfect square. That is, 26−1 = 25 = 5^2 and 26 + 1 = 27 = 3^3. Are there any other sandwich numbers? That is, can we find all integer solutions to the Diophantine equation y^3 − x^2 = 2? (I suppose my definition of sandwich numbers didn’t specify that the cube was the larger number and the square was smaller, so maybe we should also try to find solutions to y^3 − x^2 = −2 – but to keep this problem a little shorter, let’s just look at y^3 − x^2 = 2.) This problem answers the question with the use of quadratic integers!

(a) First, show that if y^3 − x^2 = 2, then y must be odd (consider the equation mod 8). From this, show that x must also be odd.
(b) Next, rewrite the equation as y^3 = x^2 + 2. The right-hand side can be factored in the quadratic field Q[√−2] as (x + √−2)(x − √−2). By Theorem 8.20 in the book, we know this is a Euclidean domain and thus a UFD. Use Theorem 8.5 (at the end of the question) to characterize the quadratic integers in Q[−2]. Also, list all the units in Q[√−2]. Is 2 prime in Q[√−2]?
(c) If x is any odd standard (rational) integer, what can be said about the greatest common factor of the quadratic integers x + √−2 and x − √−2 in Q[√−2]?
(d) If α and β are relatively prime quadratic integers in Q[√−2] and their product is a perfect cube, must α and β themselves be perfect cubes? (Note that you have already found all possible units in Q[√−2]).
(e) If x is any odd standard (rational) integer and a and b are standard (rational) integers for which x + √−2 = (a + b√−2)^3, use the binomial expansion of the right-hand side to solve for b, then a, and ultimately, x. List all possible solutions in standard (rational) integers for y^3 − x^2 = 2.

Theorem 8.5: If d is not congruent to 1 (mod 4), then the integers of Q[sqrt(d)] are exactly those numbers of the form a + b√(d), where a and b are rational integers. If d is congruent to 1 (mod 4), then the integers of Q[sqrt(d)] are those numbers of the form (a + b√(d))/2, where a and b are rational integers, both even or odd.

Added by Robert R.

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