Three in a row
(Continued from yesterday)
- Come to think of it, I do see something.
- What?
- Well I can’t explain it and it may break down further out.
- Spit out! What do you see?
- Every number has at least three zeros in a row, like 000.
- What about 1023 or 1111111111?
- Sorry, it breaks down right there.
- No reason to be sorry.
- May I change my pattern slightly?
- Be my guest!
- Every number has either 111 or 000 in them somewhere.
June 4th, 2010 at 12:24 am
x² – 1 = (x+1)(x-1).
If x is odd, x+1 and x-1 are both even. Moreover, they differ by 2, so one or other is a multiple of 4. Thus their product is a multiple of 8 and so its binary representation ends in 000.
Working modulo 8: If x=0, then x+1=1 and x-1=7. If x=4, then x+1=5 and x-1=3. In each case, (x+1)(x-1)=7 and so its binary representation ends in 111.
This leaves the case where x is even but not a multiple of 4. Working modulo 32, if x=2, 6, 10, 14, 18, 22, 26 or 30, then in each case (x+1)(x-1)=3, so its binary representation ends in 00011.
Thus in each case the binary representation of x² – 1 = (x+1)(x-1) contains 000 or 111 (and within its last 5 binary digits, at that).