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by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ.
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48:'s criterion on irrationality to the statement that a real number α is irrational if and only if there are infinitely many rational numbers
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However, Hurwitz also showed that if we omit the number φ, and numbers derived from it, then we
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173:. Again this new bound is best possible in the new setting, but this time the number
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127:{\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.}
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are a sequence of numbers that appear in bounds relating to the approximation of
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then we can increase the number on the right hand side of the inequality from 2
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477:. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 45.
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on the right hand side. The above result is best possible since the
549:
Introduction to
Diophantine methods irrationality and transcendence
201:/5. Repeating this process we get an infinite sequence of numbers
311:{\displaystyle L_{n}={\sqrt {9-{\frac {4}{{m_{n}}^{2}}}}}}
222:/5, ... which converge to 3. These numbers are called the
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This was an improvement on
Dirichlet's result which had 1/
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409:{\displaystyle m^{2}+x^{2}+y^{2}=3mxy\,}
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562:, Lagrange Numbers on pp. 24–26.
56:, written in lowest terms, such that
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419:has a solution in positive integers
180:is the problem. If we don't allow
145:φ is irrational but if we replace
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452:Conway&Guy (1996) pp.187-189
46:Peter Gustav Lejeune Dirichlet
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558:- Online lecture notes by
479:Cambridge University Press
234:Relation to Markov numbers
576:Diophantine approximation
344:such that the equation
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226:, and are named after
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32:. They are linked to
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228:Joseph Louis Lagrange
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340:th smallest integer
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159:increase the number
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505:The Book of Numbers
461:Cassels (1957) p.41
443:Cassels (1957) p.14
242:th Lagrange number
560:Michel Waldschmidt
554:2012-02-09 at the
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26:irrational numbers
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30:rational numbers
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143:golden ratio
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18:mathematics
487:0077.04801
431:References
40:Definition
539:MathWorld
499:Guy, R.K.
278:−
75:−
72:α
570:Category
552:Archived
537:. From
501:(1996).
473:(1957).
328:is the
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210:√
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321:where
20:, the
515:ISBN
423:and
238:The
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