x^2=x

[wblipp]

Maybe the PDP 11/44 handled overflow in a manner that created some additional answers? I know that an above-range integer multiplication could return a negative number - the unsigned integer result interpreted as a ones-complement or twos-complement signed integer. Perhaps it kept the lower bits.

[retina]

Originally Posted by Batalov

Ok, maybe the OP wanted to say, on a PDP 11/44, in a machine word, do some x² equal x (that is, mod 2³², for example)?

There are no other results mod 2³² either.

[TheMawn]

Originally Posted by R.D. Silverman

Even before one talks algebra one learns that you can't divide by 0.

Yes that was the idea.


I still remember the "proof" that 1 = 2. The whole idea of x = x², divide by x and get x = 1 is quite similar to the steps in that "proof". That one certainly got the gears moving. I spotted the same thing years later in some engineering lecture. Some (y - x) bracket that got divided out with nothing in particular to stop y = x cases.

[Batalov]

There were some computers that worked with packed decimal digits BCD; that would be the only opportunity.

Not on modern CPUs with 2^32 or 2^64. I violently agree!

[axn]

Another possibility: After 30 years, OP has forgotten the exact problem statement.

[LaurV]

Originally Posted by axn

Another possibility: After 30 years, OP has forgotten the exact problem statement.

hahaha
There are three signs you are getting old, one, you are forgetting things, two, you care less and less about things, and three... well... I forgot the third, but I don't care...

[LaurV]

Originally Posted by Batalov

Quote: One day in the company of friends, Hodja Nasreddin began complaining about old age.
- "However, this does not impact on my strength," - he concluded suddenly. "I am just as strong as like many years ago."
- "How do you know that?" - They asked him.
- "In my yard, there's been a huge stone. It's been there forever. So, when I was a kid, I could not pick it up; in my youth, I also could not pick it up, and I still can not pick it up now..."

This is a good one. We use to tell it differently, we say, "My hands are growing bigger and bigger as I am getting older". When I heard it first time, from a relative of mine which is about 10 years older, I was really puzzled, "What do you mean?", "Well, when I was your age I used to put my both hands on 'it' and still the head was outside, but now I only place one hand and there is nothing outside..." [edit: "there is no reminder", actually, we are on a math forum here... ]
The obvious conclusion, our hands are getting bigger as we get older...

[S485122]

Originally Posted by Batalov

Ok, maybe the OP wanted to say, on a PDP 11/44, in a machine word, do some x² equal x (that is, mod 2³², for example)?

This is akin to a perenially popular search for a ...x which squared still ends with ...x (in a certain base, e.g. in decimal) -- there are four solutions, in decimal, ...0000000, ...00000001, ...109376, and ...890625

This is indeed the what I had in mind : I programmed a routine using double integers and arrays to be able to work with large numbers. At about 10 000 digits of the last two "solutions" the computer stopped because of a problem on one of the three memory boards...

I realised after editing time was over that I should have told about the base.

Like I said I must have worded my search wrongly because I did not find a trace of this "perennial" problem : perhaps I should have searched on the last part of the "solutions".

I tried to prevent adverse comments of the serious matematicians by puting this in the puzzle sub forum and by starting to say that this was posted a bit late. Another failure ;-)

Jacob

[S485122]

I can add that by searching with part of the solutions would have conducted me to this very forum ! See the thread Playing with decimal representation .

Which shows that one (I in this case) should not post in a (sub-)forum one does not read regularly and systematically.

Sorry for having you losing time.

Jacob

[R.D. Silverman]

Originally Posted by S485122

This is indeed the what I had in mind : I programmed a routine using double integers and arrays to be able to work with large numbers. At about 10 000 digits of the last two "solutions" the computer stopped because of a problem on one of the three memory boards...

Next time, try using your brain.

Solve it mod 2, then use Hensel's Lemma.

[R.D. Silverman]

Originally Posted by R.D. Silverman

Next time, try using your brain.

Solve it mod 2, then use Hensel's Lemma.

Of course, one could also simply observe that x must be a zero divisor
of the ring Z/2^kZ and that (x-1) is its cofactor. Now ask: what are the zero divisors?