## Diophantine Analysis |

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congruence consider the equation Crelle's Journal cube Determine those Pythagorean Develop different from zero Diophantine analysis Diophantine equation Diophantine system divisible domain with respect equation x2+y2 EXERCISES exist Find a two-parameter form 4W+3 form a domain formula four numbers functional equations greatest common divisor Hence impossible in integers infinite descent infinite number integers x integral Pythagorean triangles integral squares integral value last equation lemma Mansfield Merriman means Mirimanoff mod p2 mod q number of solutions numbers whose sum Obtain solutions obvious positive integral solution prime number prime positive integers problem of finding problem of solving proof prove quadratic residue quaternions rational functions rational numbers rational triangle relation relatively prime positive respect to multiplication satisfied by integers second member second solution set of numbers Show single solution solution in integers solution of Eq solving Eq square number Thence Theory of Numbers tion two-parameter solution variables whence

### Popular passages

Page 106 - To find three squares such that the product of any two of them, added to the sum of those two, gives a square.

Page 112 - ... square of any one of them added to the next following gives a square. Let the first be x, the second 2x + i, and the third 2(2*+ i)+ i or 4*rt- 3, so that two conditions are satisfied.

Page 112 - ... and c. 35. The edges of two hollow cubes differ by 10 centimeters. If a certain quantity of water is poured into the larger cube, there remain 1578 cubic centimeters of space not filled with water. If the second cube contained 142 cubic...

Page 116 - F(x) denotes an irreducible polynomial in x of degree r (r>z) with integral coefficients and c is an integer, has only a finite number of solutions in integers p and q.

Page v - They can be got from right-angled triangles2 by dividing the square of one of the sides about the right angle by the square of the other. Let the squares then be The continued product = -J^ffax...

Page 97 - ... z, which are prime each to each, then one of these integers (say z) and the sum of the other two (say x-\-y) are both divisible by q. We shall give one other theorem which may be demonstrated by elementary means; namely, the following: V. // p is an odd prime and the equation...

Page 115 - ... the product of two sums of eight squares as a sum of eight squares.