Exact values of the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>Γ</mml:mi><mml:mo>⁎</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

For [Formula: see text] and [Formula: see text] a prime number, define [Formula: see text] to be the smallest positive integer [Formula: see text] such that any diagonal form [Formula: see text], with integer coefficients, has nontrivial zero over [Formula: see text] whenever [Formula: see text]. A special case of a conjecture attributed to Artin states that [Formula: see text]. It is well known that the equality occurs when [Formula: see text]. In this paper, we obtain the exact values of [Formula: see text] for all primes [Formula: see text] and, except for [Formula: see text], these values are much lower than those established in the conjecture, as might be expected.

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Exact values of the function Γ⁎(k)

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References 7 publications

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