Frobenius numbers of Pythagorean triples
نویسندگان
چکیده
منابع مشابه
Pythagorean Triples
Let n be a number. We say that n is square if and only if: (Def. 3) There exists m such that n = m2. Let us note that every number which is square is also natural. Let n be a natural number. Note that n2 is square. Let us observe that there exists a natural number which is even and square. Let us observe that there exists a natural number which is odd and square. Let us mention that there exist...
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The name comes from elementary geometry: if a right triangle has leg lengths x and y and hypotenuse length z, then x + y = z. Of course here x, y, z are positive real numbers. For most integer values of x and y, the integer x + y will not be a perfect square, so the positive real number √ x2 + y2 will be irrational: e.g. x = y = 1 =⇒ z = √ 2. However, a few integer solutions to x + y = z are fa...
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then the Fibonacci numbers are given by Fn = Fn (1), and the Pell numbers are Pn = Fn(2). Cohn [4] has proved that the only perfect squares among the sequences {Fn(a)}9 a odd, are 0 and 1, and whenever a = k, a itself. Certain cases are known for a even [5]. The cited results of Cohn and Ljunggren mean that Conjectures 2.3,3.2, and 4.2 of [3] are true, and that the earlier results can be streng...
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We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite, and is equal to the address of the PPT on Barning’s [9] ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugat...
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A Pythagorean triple is a triple of positive integers a,b,c ∈ N+ satisfying a2 + b2 = c2. Is it true that, for any finite coloring of N+, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X2 +Y 2 = Z2 regular? This problem has been open since several decades, even restricted to 2-colorings. In this note, we introduce partial morphisms, which are ...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2015
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042115500323