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forum.math.uoa.gr • Προβολή θέματος - Σύνολο φυσικών

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 Θέμα δημοσίευσης: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 24 Σεπ 2016, 02:23 
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Εγγραφή: 05 Φεβ 2008, 03:03
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Έστω S υποσύνολο των φυσικών αριθμών, τέτοιο ώστε το [unparseable or potentially dangerous latex formula] να είναι άπειρο. Υποθέτουμε επίσης ότι το S είναι κλειστό ως προς την πρόσθεση: αν [unparseable or potentially dangerous latex formula], τότε [unparseable or potentially dangerous latex formula]. Δείξτε ότι υπάρχει φυσικός n\geq 2 ο οποίος να διαιρεί κάθε στοιχείο του S.

_________________
\emptyset\not=\{\emptyset\}


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 28 Σεπ 2016, 08:34 
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Άβαταρ μέλους

Εγγραφή: 25 Φεβ 2008, 12:08
Δημοσ.: 78
Μιά γρήγορη προσέγγιση:

Άν 1\in S τότε [unparseable or potentially dangerous latex formula] επομένως το [unparseable or potentially dangerous latex formula].
Έστω 1\notin S και [unparseable or potentially dangerous latex formula] με [unparseable or potentially dangerous latex formula] τότε [unparseable or potentially dangerous latex formula].
Τέλος αν 1\notin S και a,b \in S με a<b και gcd(a,b)=1 τότε [unparseable or potentially dangerous latex formula].
Αρκεί να δείξουμε πως υπάρχουν επίσης και οι [unparseable or potentially dangerous latex formula]. Τότε θα έχουμε δημιουργήσει a διαδοχικούς όρους που παράγουν όλους τους (επόμενους από αυτούς) φυσικούς. Τότε το [unparseable or potentially dangerous latex formula] και [unparseable or potentially dangerous latex formula] πεπερασμένο.

Ψάχνω λοιπόν x,y \in \mathbb N ώστε [unparseable or potentially dangerous latex formula]. Επειδή gcd(a,b)=1 , οι διοφαντικές αυτές δίνουν λύση.
μένει να ελένξουμε αν έχουν λύση στους φυσικούς..

_________________
Σ'ένα άγριο όνειρο ξύπνησα παγωμένος,
κάποιος με ρώταγε μέσα από το μαύρο φώς:
Ποιός είναι ο δρόμος μου,αν ξέρω που πηγαίνω,
κι αν ξέρω ποιος από τους δυό μας είναι αυτός,
και ποιός εγώ;


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 28 Σεπ 2016, 08:41 
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Εγγραφή: 05 Φεβ 2008, 03:03
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graps έγραψε:
Έστω 1\notin S και [unparseable or potentially dangerous latex formula] με [unparseable or potentially dangerous latex formula] τότε [unparseable or potentially dangerous latex formula].


Εδώ το d είναι σταθερό, ή εξαρτάται από τα a, b;
Αν υπάρχει d τέτοιο ώστε, για κάθε [unparseable or potentially dangerous latex formula], τότε [unparseable or potentially dangerous latex formula], αυτό συνεπάγεται ότι όλα τα στοιχεία του S είναι πολλαπλάσια του d. Αν όμως δεν υπάρχει τέτοια ομοιόμορφη επιλογή για το d?

_________________
\emptyset\not=\{\emptyset\}


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 28 Σεπ 2016, 08:46 
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Άβαταρ μέλους

Εγγραφή: 25 Φεβ 2008, 12:08
Δημοσ.: 78
καταρχάς το θεώρησα σταθερό, αν πάλι δεν είναι και υπάρχουν διαφορετικά gcd ανά δύο ας πούμε, τότε η περίπτωση αυτή εμπίπτει στον μικρότερο d που διαιρεί τους υπόλοιπους, αν δεν υπάρχει τέτοιος καθολικά τότε θα υπάρχουν τουλάχιστον 2 d με [unparseable or potentially dangerous latex formula]

_________________
Σ'ένα άγριο όνειρο ξύπνησα παγωμένος,
κάποιος με ρώταγε μέσα από το μαύρο φώς:
Ποιός είναι ο δρόμος μου,αν ξέρω που πηγαίνω,
κι αν ξέρω ποιος από τους δυό μας είναι αυτός,
και ποιός εγώ;


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 28 Σεπ 2016, 08:49 
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Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
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Σωστά. Άρα όντως ανάγεται, προς άτοπο, στο ότι ο μέγιστος κοινός διαιρέτης όλων των x\in S είναι ίσος με 1.

Η ιδέα με τους διαδοχικούς όρους δουλεύει, θέλει όμως μια προσαρμογή.

_________________
\emptyset\not=\{\emptyset\}


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 28 Σεπ 2016, 08:51 
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Άβαταρ μέλους

Εγγραφή: 25 Φεβ 2008, 12:08
Δημοσ.: 78
τέλεια. Ρίξε άλλη προσέγγιση αν έχεις για να δω άλλες σκέψεις πάνω στο πρόβλημα αν είχες σκεφτεί κάτι διαφορετικό.

_________________
Σ'ένα άγριο όνειρο ξύπνησα παγωμένος,
κάποιος με ρώταγε μέσα από το μαύρο φώς:
Ποιός είναι ο δρόμος μου,αν ξέρω που πηγαίνω,
κι αν ξέρω ποιος από τους δυό μας είναι αυτός,
και ποιός εγώ;


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 Θέμα δημοσίευσης: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 30 Σεπ 2016, 20:37 
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Εγγραφή: 11 Φεβ 2007, 21:13
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Αυτό είναι το Frobenius problem…NP -hard… Είναι γνωστό και ως coin problem

Εδώ σελίδα 96 πχ

https://www.math.upenn.edu/~wilf/gfology2.pdf

και εδώ σελίδα 53

http://www.fmf.uni-lj.si/~lavric/Santos%20-%20Number%20Theory%20for%20Mathematical%20Contests.pdf


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 01 Οκτ 2016, 06:59 
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Εγγραφή: 05 Φεβ 2008, 03:03
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Για το γενικό πρόβλημα, ναι. Εδώ όμως δεν είναι απαραίτητο να χρησιμοποιηθεί.

Σαν υπόδειξη, υπάρχουν δύο διαδοχικοί φυσικοί που είναι στοιχεία του συνόλου (γιατί;). Χρησιμοποιόντας αυτούς τους δύο, μπορούμε να κατασκευάσουμε [unparseable or potentially dangerous latex formula] (ποιον;) τέτοιο ώστε, αν m\geq n, τότε m\in S.

_________________
\emptyset\not=\{\emptyset\}


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 Θέμα δημοσίευσης: Re: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 01 Οκτ 2016, 12:38 
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Εγγραφή: 29 Οκτ 2010, 01:19
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graps έγραψε:
καταρχάς το θεώρησα σταθερό, αν πάλι δεν είναι και υπάρχουν διαφορετικά gcd ανά δύο ας πούμε, τότε η περίπτωση αυτή εμπίπτει στον μικρότερο d που διαιρεί τους υπόλοιπους, αν δεν υπάρχει τέτοιος καθολικά τότε θα υπάρχουν τουλάχιστον 2 d με [unparseable or potentially dangerous latex formula]


Έστω [unparseable or potentially dangerous latex formula] με gcd(a,b)=1.
Τότε, επειδή ο a είναι γεννήτορας της προσθετικής ομάδας [unparseable or potentially dangerous latex formula] για μεγάλο k\in \mathbb{N} έχουμε [unparseable or potentially dangerous latex formula]. // το [unparseable or potentially dangerous latex formula] πιάνεται απο κάποιο πολλαπλάσιο του a.

Άρα υπάρχουν διαδοχικοί φυσικοί [unparseable or potentially dangerous latex formula] στο S.
Τώρα έχουμε:

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
...
[unparseable or potentially dangerous latex formula]

κάθε φορά αφαιρώ ένα n και προσθέτω ένα n+1 άρα προχωράω +1 στην ευθεία των αριθμών μέχρι να φτάσω τελικά στο επόμενο πολλαπλάσιο του n.
Ξεκίνησα με τόσα n ώστε να με βγάλουν στη διαδρομή μου ([unparseable or potentially dangerous latex formula]).
Στο εξής, θα χω και περίσσευμα.

Έτσι πηγαίνοντας απ το ένα πολλαπλάσιο του n στο επόμενο συμπεραίνουμε οτι όλοι οι φυσικοί απ το n^2 και μετά ανήκουν στοS.
Άτοπο, καθώς τότε το [unparseable or potentially dangerous latex formula] είναι άπειρο.


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 Θέμα δημοσίευσης: Σύνολο φυσικών
ΔημοσίευσηΔημοσιεύτηκε: 05 Οκτ 2016, 00:54 
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Εγγραφή: 11 Φεβ 2007, 21:13
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Υπάρχουν ἀκέραιοι αριθμοί [unparseable or potentially dangerous latex formula] τέτοιοι ώστε [unparseable or potentially dangerous latex formula]
Αυτὀ εἰναι το γενικευμένο θεώρημα του Bezout (που είναι NP-complete)

Έστω \Sigma το σύνολο της μορφής αυτής. Η Αρχή τής καλής διάταξης μάς εξασφαλίζει ότι το σύνολο έχει ένα ελάχιστο στοιχείο, το d
Έστω ότι:
[unparseable or potentially dangerous latex formula]

Θα αποδείξουμε ότι αυτό το d είναι ο ΜΚΔ τών [unparseable or potentially dangerous latex formula]
Απο τον ευκλείδιο αλγόριθμο έχουμε
[unparseable or potentially dangerous latex formula]
Με [unparseable or potentially dangerous latex formula] και :
[unparseable or potentially dangerous latex formula]=
[unparseable or potentially dangerous latex formula], που αντιβαίνει την υπόθεση περι ελαχίστου στοιχείου. Άρα [unparseable or potentially dangerous latex formula] Για κάθε [unparseable or potentially dangerous latex formula] έχουμε τότε [unparseable or potentially dangerous latex formula] οπότε [unparseable or potentially dangerous latex formula] που καταδεικνύει τήν ύπαρξη των [unparseable or potentially dangerous latex formula]

Τώρα, μάς ενδιαφέρει η περίπτωση που d=1 Όμως αυτό είναι σαν να λέμε ότι το S είναι ένα υπομονοειδές (αριθμητική ημιομάδα) τού \mathbb{N} πράγμα αδύνατον λόγω τού ότι το συμπλήρωμα τού S είναι άπειρον.


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