 mersenneforum.org > Math factors of surds
 Register FAQ Search Today's Posts Mark Forums Read 2018-04-02, 11:32 #1 wildrabbitt   Jul 2014 19×23 Posts factors of surds Hello, 446 + 177 = (7 + 11)*(4 + 19) I picked some numbers and calculated the LHS. That's why I can write it the other way around. Can someone tell what kind of maths I need to use to work from the left to the right. i.e Given a + b, how can I work out factors of the same form? Last fiddled with by science_man_88 on 2018-04-02 at 12:22   2018-04-02, 11:42   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts Quote:
 Originally Posted by wildrabbitt Hello, 446 + 177 = (7 + 11)*(4 + 19) I picked some numbers and calculated the LHS. That's why I can write it the other way around. Can someone tell what kind of maths I need to use to work from the left to the right. i.e Given a + b, how can I work out factors of the same form?
Knowing (a+b)(c+d)= ac+bc+ad+bd may help. Edit: ex. In your own example 146= 7*4+11*19*2, the two came out divide 146 by 2 you get 73, that means you have one product even one product odd in what's left, namely 7*2 and 11*19. 177√2 is 7*19√2+4*11√2 which again points to an even odd split.

Last fiddled with by science_man_88 on 2018-04-02 at 12:22   2018-04-02, 12:20 #3 Nick   Dec 2012 The Netherlands 26·3·7 Posts If $$x=a+b\sqrt{2}$$, define $$N(x)=a^2-2b^2$$. Then for all x and y we have $$N(xy)=N(x)N(y)$$ so looking at the absolute value of $$N(x)$$ enables you to determine a finite list of possible factors of $$x$$ You can find out more about this by looking up quadratic fields in a good book on Algebraic Number Theory, for example: https://www.crcpress.com/Algebraic-N.../9781498738392   2018-04-02, 13:29 #4 wildrabbitt   Jul 2014 19·23 Posts Thanks to both posters.   2018-09-30, 18:02 #5 wildrabbitt   Jul 2014 43710 Posts Hi again, does anyknow if this is true implies & ?   2018-10-01, 09:00   #6
Nick

Dec 2012
The Netherlands

26·3·7 Posts Quote:
 Originally Posted by wildrabbitt Hi again, does anyknow if this is true implies & ?
No - for example, put a=7, b=0, c=e=3, d=f=1.   2018-10-01, 09:17 #7 wildrabbitt   Jul 2014 19·23 Posts thanks   2018-10-01, 13:42 #8 Dr Sardonicus   Feb 2017 Nowhere 57238 Posts In addition to Nick's example, based on primes of the form a^2 - 2*b^2 [namely p == 1 or 7 (mod 8)], a "trivial" class of counterexamples uses "units of norm 1"; that is, solutions to e^2 - 2*f^2 = 1, e.g. e = 3, f, = 2. We can take c = a, d = b, and e = 3, f = 2 [or c = a, d = b, and e = 17, f = 12; or c = a, d = b, and e = 99, f = 70, etc]   2018-10-01, 17:28 #9 wildrabbitt   Jul 2014 1101101012 Posts Thanks.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post ATH PrimeNet 2 2014-09-04 19:45 MatWur-S530113 PrimeNet 11 2009-01-21 19:08 ATH Prime Cullen Prime 16 2007-07-07 13:02 MatWur-S530113 Math 21 2007-05-12 19:36 dave_0273 Marin's Mersenne-aries 5 2004-12-24 12:54

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