What do these numbers have in common?

41, 80 and 320?

Soroban

MHF Hall of Honor

Um . . . Their binary representations begin with "101" ?

. . $$\displaystyle \begin{array}{ccc} 41 &=& 101001_2 \\ 80 &=& 1010000_2 \\ 320 &=& 101000000_2 \end{array}$$

bigblackbronco

they can all be / by 1

Soroban

MHF Hall of Honor

Don't know how I missed that . . .

bigblackbronco said:
they can all be / by 1

And I wasted all that time looking for patterns . . . *sigh*

. . $$\displaystyle \begin{array}{ccc} 41 &=& 4^2 + 5^2 \\ 80 &=& 4^2 + 8^2 \\ 320 &=& 8^2 + 16^2 \end{array}$$

eumyang

Keep going.

eumyang

If you add the three numbers you get a perfect square:
$$\displaystyle 41 + 80 + 320 = 441 = 21^2$$

And... if you add any two of the numbers you also get perfect squares:
\displaystyle \begin{aligned} 41 + 80 &= 121 = 11^2 \\ 80 + 320 &= 400 = 20^2 \\ 41 + 320 &= 361 = 19^2 \end{aligned}

If it wasn't for Soroban's last post I wouldn't have thought of this. (Bow)

wonderboy1953

If you add the three numbers you get a perfect square:
$$\displaystyle 41 + 80 + 320 = 441 = 21^2$$

And... if you add any two of the numbers you also get perfect squares:
\displaystyle \begin{aligned} 41 + 80 &= 121 = 11^2 \\ 80 + 320 &= 400 = 20^2 \\ 41 + 320 &= 361 = 19^2 \end{aligned}

If it wasn't for Soroban's last post I wouldn't have thought of this. (Bow)
That's the answer I was looking for (I wonder how this plays out in binary?)

Chokfull

That's the answer I was looking for (I wonder how this plays out in binary?)
It shouldn't change a bit in binary as this solution has nothing to do with the numbers' digits, only their values.