Euclids Classification of Pythagorean triples

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Euclids Classification of Pythagorean triples

• Integer solutions to x2y2z2. These give right
  triangles with integer sides.
• Examples (2n1,2n22n,2n22n1) and
  (2n,n2-1,n21).
• The concept of similar plane numbers a plane
  number is simply a formal product of two numbers,
  thought of as the sides of a rectangle, of which
  the plane number itself gives the area.
• Two plane numbers are similar if their respective
  sides are proportional with the same rational
  proportionality factor.
• Euclid, from Elements
• LEMMA 1 (before Proposition 29 in Book X) To
  find two square numbers such that their sum is
  also a square.
• From similar plane numbers, one can construct
  Pythagorean triples.
• Moreover all P. triples arise in this way!
• Lemma Two plane numbers are similar if and only
  if their ratio is the square of a rational
  number.
• All primitive triples are of the form
• dp2-q2, e2pq, fp2q2, with p and q relatively
  prime.

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Euclids Classification of Pythagorean triples

• LEMMA 1 (before Proposition 29 in Book X) To
  find two square numbers such that their sum is
  also a square.
• Let two numbers AB, BC be set out, and let them
  be either both even or both odd. Then since,
  whether an even number is subtracted from an even
  number, or an odd number from an odd number, the
  remainder is even IX. 24, 26, therefore the
  remainder AC is even. Let AC be bisected at D.
  Let AB, BC also be either similar plane numbers,
  or square numbers, which are themselves also
  similar plane numbers. Now the product of AB, BC
  together with the square on CD is equal to the
  square on BD II. 6. And the product of AB, BC
  is square, inasmuch as it was proved that, if two
  similar plane numbers by multiplying one another
  make some number the product is square IX. 1.
  Therefore two square numbers, the product of AB,
  BC, and the square on CD, have been found which,
  when added together, make the square on BD. And
  it is manifest that two square numbers, the
  square on BD and the square on CD, have again
  been found such that their difference, the
  product of AB, BC, is a square, whenever AB, BC
  are similar plane numbers. But when they are not
  similar plane numbers, two square numbers, the
  square on BD and the square on DC, have been
  found such that their difference, the product of
  AB, BC, is not square.

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Euclids Classification of Pythagorean triples

• Start with d2e2f2 with d, e and f co-prime and
  suppose that e is even.
• f2-d2e2uv with ufd and vf-d
• So f(uv)/2 and d(u-v)/2.
• For uv to be a square they have to be similar
  plane numbers by Euclid. They also have to have
  of the same parity in the case above even.
• By the lemma they are of the form ump2 and vmq2
  with co-prime p and q and moreover m2 and p and
  q have different parity. This yields all
  primitive triples. In the form u2p2, v2q2 and
  so
• dp2-q2, e2pq, fp2q2
• with p, q, pgtq relatively prime and of different
  parity.

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Proof of the Lemma

• Lemma Two plane numbers are similar if and only
  if their ratio is the square of a rational
  number.
• Proof. If uxy and vzw are similar plane
  numbers, then yrx and zrw for some r in Q and
• u/vxy/zwry2/rw2(y/w)2.
• If u/v (p/q)2p2/q2, with coprime p,q then u
  p2/q2v and hence q2 divides v and vmq2. Also v
  q2/p2v and hence p2 divides v and wvp2. But
  u/v mp2/(mq2)p2/q2, so mn. And u(mp)p,
  v(mq)q are similar plane numbers.