20100528, 12:46  #1 
Banned
"Luigi"
Aug 2002
Team Italia
3×5×317 Posts 
Square numbers and binary representation
I just noticed (after reading a book from Roberto Vacca) that a perfect square number turned into binary form has the second digit from right always equal to zero.
Code:
2^{2} = 4 = 100 3^{2} = 9 = 1001 4^{2} = 16 = 10000 5^{2} = 25 = 11001 6^{2} = 36 = 100100 7^{2} = 49 = 110001 8^{2} = 64 = 1000000 9^{2} = 81 = 1010001 10^{2} = 100 = 1100100 11^{2} = 121 = 1111001 13^{2} = 169 = 10101001 19^{2} = 361 = 101101001 I know there should be a theorem in Number Theory (published in Computational Mathematics, IIRC) related to statistical presence of "zeroes" and "ones" in binary representation of square numbers. Any hints? Luigi 
20100528, 12:59  #2  
Nov 2003
2·3·11·113 Posts 
Quote:
And this is all there is to it. Quote:


20100528, 13:41  #3 
Feb 2006
Denmark
2×5×23 Posts 
All odd squares are 1 mod 4. All even squares are 0 mod 4. In either case the second bit from the right is 0. ET gave a correct argument for that. He may not have formulated it the same way a mathematician typically would and it's a trivial observation but I don't think it's fair to say "No" to him.

20100528, 13:51  #4  
Nov 2003
2×3×11×113 Posts 
Quote:
forum title. 

20100528, 14:06  #5 
"(^r'°:.:)^n;e'e"
Nov 2008
;t:.:;^
3^{3}×37 Posts 
to Luis :
s..pari ..°° di(s)_im_pari ..°' ( it ) ghe duma (a_mo) do posibilità e do_po dem via ... in un alter "pais" magari del quart mund '° ... dan_y ?! Last fiddled with by cmd on 20100528 at 14:57 Reason: it 
20100528, 14:21  #6  
Feb 2006
Denmark
2×5×23 Posts 
Quote:
ET's post was mathematical regardless of formulation and it seemed more correct than your reply. You said "No" to a valid argument and forgot to include even squares in your own argument. Your posts usually have a high mathematical level and I'm sure you are better at mathematics than ET and me. I just think you were a little too fast in this case. 

20100528, 15:17  #7  
"(^r'°:.:)^n;e'e"
Nov 2008
;t:.:;^
3^{3}×37 Posts 
ps it
Quote:
miiET city ( no tifosi ) 

20100528, 18:50  #8  
Nov 2003
2×3×11×113 Posts 
Quote:
posting here are incompetent? I would think that they would find that insulting. Is there something wrong in having an expectation of competency? 

20100528, 19:01  #9 
"(^r'°:.:)^n;e'e"
Nov 2008
;t:.:;^
999_{10} Posts 
@ET_
cmd=crapa"sempre+pelada" ... solo l'ultimo minuto ti può interessare ... il resto è no_ia, ( segui bene il foglio e nota il grafico ) vedi i colori rosa ed azzurro ?! Last fiddled with by cmd on 20100528 at 19:17 Reason: A.G.G.itur 
20100528, 21:18  #10 
Feb 2006
Denmark
346_{8} Posts 
No, but I would sometimes prefer a change in attitude towards people who don't live up to your expectations. This is an Internet forum, not an advanced math class. Experience indicates that when you react to mathematical shortcomings in posters, the thread often goes downhill. It happens more rarely when others react to the same shortcomings in a different tone. You may blame incompetent posters for it but if a bad pattern persists then you can argue about whose fault it is, or you can try to change the pattern.

20100528, 21:30  #11  
Nov 2003
2·3·11·113 Posts 
Quote:
subject. It is also not a subject for those who lack mathematical maturity. It is a subject for those who have mastered (at least) secondary school algebra. 

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