Make 113 using four 4s and ANY mathematical operations or functions?

Question

The only numbers or integers allowed in the expression are four 4s, and all functions must be differentiable functions, i.e., continuous functions.

Doctor Q, you've used 2 for an exponential. Mamad, you're using eight 4s, not just four 4s.

Also, the expression cannot be infinitely recursive.

Furthermore, you can't use "0" and claim it's not an number. It is a number.

Folks, you cannot use "2", and you cannot use "0"!

Another version of Mathemagician's solution would be (my version):

Γ(4+√4) - Γ(4) - Γ(√4) = 113

Mathemagician's right, the gamma function Γ(x) is continuously differentiable.

Answer 1

You can make 113 using the gamma function (Γ). The gamma function is defined for integers by

Γ(n + 1) = n!

So, in particular, we have

Γ(4) = 3! = 6, and Γ(Γ(4)) = Γ(6) = 5! = 120.

Then

Γ(Γ(4)) - (4! + 4)/4 = 113.

(The gamma function is not usually encountered before a course in analysis, but it is a very important and well-known function in advanced mathematics.)

The number 113 is (to the best of my knowledge) impossible to obtain using just addition, subtraction, multiplication, division, exponents, square roots, and factorials. You need to introduce something else. The gamma function is a well-known function you can introduce to make 113 without being too cheap.

Oh, and Γ is differentiable at all positive numbers. (It's defined for non-integers also; you can look up the gamma function on Wikipedia if you are unfamiliar with it and want to know more about it.)

Answer 2

4 x 4! + 4^2 + 4^0 = 113

Answer 3

I don't suppose this is of
any use, but here goes -
(Note decimal point)
113 = INT[(4!*4 - 4!) / √(.4)]

EDIT:
Then there's 4^4 * .44
= 112.64 to 2 decimal places
= 112.6 to 1 decimal place
= 113 to 0 decimal places

I know, I know! I'm still thinking.

Answer 4

hi
my ans is
4*4!+4^2+4^0=113

Answer 5

Does this count?

(4! + 4) * 4 + ceiling(cos(4))

Answer 6

44*4-4*4*4+4/4
44*4=176
4*4*4=64
4/4=1
176-64+1=113