Square numbers and binary representation

[ET_]

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.

 22 =   4 = 100
 32 =   9 = 1001
 42 =  16 = 10000
 52 =  25 =  11001
 62 =  36 = 100100
 72 =  49 = 110001
 82 =  64 = 1000000
 92 =  81 = 1010001
102 = 100 = 1100100
112 = 121 = 1111001
132 = 169 = 10101001
192 = 361 = 101101001

This depends on the fact that during multiplication, the four possibilities (00x00, 01x01, 10x10, 11,11) always bring to a "0" in the second place from right.

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

[R.D. Silverman]

Originally Posted by ET_

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.

This depends on the fact that during multiplication, the four possibilities (00x00, 01x01, 10x10, 11,11) always bring to a "0" in the second place from right.

No. What is really happening is quite simple. All squares are 1 mod 4.
And this is all there is to it.

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.

What do you mean by "statistical presence"?? This is not well defined.

[Jens K Andersen]

Originally Posted by R.D. Silverman

No. What is really happening is quite simple. All squares are 1 mod 4.
And this is all there is to it.

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.

[R.D. Silverman]

Originally Posted by Jens K Andersen

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.

Is this not a sub-forum for *mathematical* discussion? Look at the
forum title.

[cmd]

to Luis :

s..pari ..°°
di(s)_im_pari ..°'


( it ) ghe duma (a_mo) do po-si-bi-li-tà e do_po dem via ... in un alter "pais" magari del quart mund


'° ... dan_y

?!

[Jens K Andersen]

Originally Posted by R.D. Silverman

Is this not a sub-forum for *mathematical* discussion? Look at the forum title.

I don't know whether it was first posted here or moved but I saw it in Miscellaneous Math Threads which doesn't have high math expectations (although you appear to generally have higher expectations of posters than anybody else at all of mersenneforum.org).
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.

[cmd]

Originally Posted by cmd

to Luis :

s..pari ..°°
di(s)_im_pari ..°'


( it ) ghe duma (a_mo) do po-si-bi-li-tà e do_po dem via ... in un alter "pais" magari del quart mund


'° ... dan_y

?!

... canta che ti passa ...

miiET cit-y



( no tifosi )

[R.D. Silverman]

Originally Posted by Jens K Andersen

(although you appear to generally have higher expectations of posters than anybody else at all of mersenneforum.org).
.

Would you prefer that I change my expectations to assume that people
posting here are incompetent? I would think that they would find that
insulting.

Is there something wrong in having an expectation of competency?

[cmd]

@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 ?!

[Jens K Andersen]

Originally Posted by R.D. Silverman

Would you prefer that I change my expectations to assume that people
posting here are incompetent?

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.

[R.D. Silverman]

Originally Posted by Jens K Andersen

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..

Computational number theory is a moderately advanced mathematical
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.