# Most consecutive positive integers using two 1s

Using two 1s, try to come up with the most consecutive positive integers.

Allowed operations:

• Subtraction

• Multiplication

• Division

• Concatenation

• Square Root

• Factorial

• Floor and Ceiling Functions

• Decimal Point

• Isn't there a conjecture that with only factorial, square root and floor, and starting with any single number >1, you can form all positive integers? If so, this seems like one that could go on for ever... Jan 23 '17 at 3:52
• @GarethMcCaughan It'd have to be $>2$, because $2!=2$, floor decreases and square root makes closer to 1. Jan 23 '17 at 4:34
• @GarethMcCaughan ... and floor yields an integer. Jan 23 '17 at 7:04
• Oops, yes, I meant >2. Jan 23 '17 at 10:04
• Do you happen to recall the name of this conjecture? Jan 24 '17 at 0:48

Let's try (feel free to add on or correct, this is community wiki):

$$1=1\times1$$

$$2=1+1$$

$$3=\left\lfloor\sqrt{11}\right\rfloor$$

$$4=\left\lceil\sqrt{11}\right\rceil$$

$$5=\left\lfloor\sqrt{\sqrt{\left(\left\lfloor\sqrt{11}\right\rfloor!\right)!}}\right\rfloor$$

$$6=\left\lfloor\sqrt{11}\right\rfloor!$$

$$7=\left\lceil\sqrt{\sqrt{11}!}!\right\rceil$$

$$8=\left\lfloor\sqrt{\sqrt{\sqrt{11!}}}\right\rfloor$$

$$9=\left\lfloor\sqrt{11}!\right\rfloor$$

$$10=\left\lceil\sqrt{11}!\right\rceil$$

$$11=11$$
$$12=\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{(\left\lceil\sqrt{11}\space\right\rceil!)!}}}}\right\rceil!}}}}}\right\rceil$$
$$13=\left\lceil\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{\sqrt{(\left\lceil\sqrt{11}\space\right\rceil!)!}}}}\right\rceil!}}}}}\right\rceil}}}\right\rceil$$

$$14=\sqrt{\sqrt{\left\lfloor\sqrt{\sqrt{\left\lceil\sqrt{\sqrt{\sqrt{(1\div(.1))!}}}\right\rceil!}}\right\rfloor!}}$$

• I found a way for 12 but it's just a hassle to write it down. We can go all the way to infinity without much problem. Jan 23 '17 at 6:05
• Getting past 10 was the hard part.
– humn
Jan 23 '17 at 6:08
• @humn I simplified my 10 as well as the 7 and 9. Looks much better now. But I guess 11 is a good number to stop this... I am not going any higher lol. Jan 23 '17 at 6:17
• Anyone want to write a program for this...? Jan 23 '17 at 6:19
• This is turning into a ridiculous open-ended mathematics exercise. It seems likely that given an unlimited number of factorials, square roots, floor and ceilings, you could express the values of all positive integers. Jan 23 '17 at 13:46