Relativ Prim < Zahlentheorie < Algebra+Zahlentheo. < Hochschule < Mathe < Vorhilfe
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(Frage) beantwortet  | | Datum: | 09:37 So 03.03.2013 | | Autor: | sissile |
| Aufgabe | Given a positive integer k, we can find infinitely many positive
integers a such that the k integers in the set
a + 1 , 2a + 1 , . . . , ka + 1
are pairwise relatively prime.
Proof. Let a be any positive integer which is divisible by all of the prime numbers
which are less than k. We claim that a + 1, 2a + 1, . . . , ka + 1 are pairwise
relatively prime. Suppose not. Let i and j be such that 1 ≤ i < j ≤ k and ia+1
and ja+1 are not relatively prime. Let p be a prime number which is a factor of both ia+1 and ja+1. Then p cannot be a factor of m. Hence p is greater than or equal to k. On the other hand p is a factor of (ja + 1) − (ia + 1) = (j − i)a.
Hence p is a factor of j − i. But j − i is less than k, hence p is less than k, a
Widerspruch. |
Hallo
Was soll m sein? . Denn genau die Stelle verstehe ich nicht, dass p [mm] \ge [/mm] k sein muss.
Hat da wer eine Idee, wwarum das gilt?
Liebe grüße
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(Antwort) fertig  | | Datum: | 09:54 So 03.03.2013 | | Autor: | felixf |
Moin!
> Given a positive integer k, we can find infinitely many
> positive
> integers a such that the k integers in the set
> a + 1 , 2a + 1 , . . . , ka + 1
> are pairwise relatively prime.
>
> Proof. Let a be any positive integer which is divisible by
> all of the prime numbers
> which are less than k. We claim that a + 1, 2a + 1, . . .
> , ka + 1 are pairwise
> relatively prime. Suppose not. Let i and j be such that 1
> ≤ i < j ≤ k and ia+1
> and ja+1 are not relatively prime. Let p be a prime number
> which is a factor of both ia+1 and ja+1. Then p cannot be a
> factor of m. Hence p is greater than or equal to k. On the
> other hand p is a factor of (ja + 1) − (ia + 1) = (j −
> i)a.
> Hence p is a factor of j − i. But j − i is less than
> k, hence p is less than k, a
> Widerspruch.
>
> Hallo
> Was soll m sein?
Wenn $m = a$ sein soll (ist vermutlich so gemeint gewesen), dann stimmt alles.
LG Felix
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(Frage) beantwortet | | Datum: | 10:07 So 03.03.2013 | | Autor: | sissile |
Hallo,
ok -> p teilt nicht a
Denn wenn p teilt ia+1 und p teil a so würde p die 1 teilen .
danke
LG
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Hallo sissile,
ist das eine Frage? Oder hast Du Dich nur verklickt?
> ok -> p teilt nicht a
> Denn wenn p teilt ia+1 und p teil a so würde p die 1
> teilen .
Jawollja. Genau. So isses.
Grüße
reverend
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