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In additive number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers is generated by a linear recurrence relation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. More precisely, this set of positions can be decomposed into the union of a finite set and finitely many full arithmetic progressions. Here, an infinite arithmetic progression is full if there exist integers ''a'' and ''b'' such that the progression consists of all positive integers equal to ''b'' modulo ''a''. This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0). Its proofs use p-adic analysis. ==Example== Consider the sequence :0, 0, 1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, ... that alternates between zeros and the Fibonacci numbers. This sequence can be generated by the linear recurrence relation : (a modified form of the Fibonacci recurrence), starting from the base cases ''F''(1) = ''F''(2) = ''F''(4) = 0 and ''F''(3) = 1. For this sequence, ''F''(''i'') = 0 if and only if ''i'' is either one or even. Thus, the positions at which the sequence is zero can be partitioned into a finite set (the singleton set ) and a full arithmetic progression (the positive even numbers). In this example, only one arithmetic progression was needed, but other recurrence sequences may have zeros at positions forming multiple arithmetic progressions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Skolem–Mahler–Lech theorem」の詳細全文を読む スポンサード リンク
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