# Show 2018th term of Sylvester's Sequence isn't a perfect square

A sequence $$\displaystyle T_1, T_2, T_3 ...$$ is defined by:
$$\displaystyle T_1 = 1$$
$$\displaystyle T_2 = 2$$
$$\displaystyle T_{n+1} = 1 + \prod_{i=1}^{n} T_{i}$$ for all integers $$\displaystyle n \geq 2$$

Prove that $$\displaystyle T_{2018}$$ is not a perfect square.

For context, this was given to me by a student, and is the last part of a four part question. The previous parts were:

(a) What is the value of $$\displaystyle T_5$$? (Ans: 1 + 1*2*3*7 = 43)
(b) Prove that $$\displaystyle T_{n+1} = T_n^2 - T_n + 1$$ for $$\displaystyle n \geq 2$$
(Ans:
Observe that $$\displaystyle T_n^2 - T_n = T_n(T_n - 1) = T_n($$$$\displaystyle \prod_{i=1}^{n-1} T_{i}) = T_{n+1} - 1$$
then add one on both sides)

(c) Prove that $$\displaystyle T_n + T_{n+1}$$ is a factor of $$\displaystyle T_nT_{n+1} - 1$$ for all integers $$\displaystyle n \geq 2$$
(Ans:
$$\displaystyle T_nT_{n+1} - 1 = T_n^3 - T_n^2 + T_n - 1 = (T_n^2 + 1)(T_n - 1) = (T_n + T_{n+1})(T_n - 1)$$

On part (d), which is to prove that $$\displaystyle T_{2018}$$ is not a perfect square, I'm stumped. How might one approach this? I might be missing something obvious as the structure of the question leads me to believe (c) is a fact I can use.
Either I can show the square root of a representation of $$\displaystyle T_{2018}$$ isn't rational, or show $$\displaystyle T_{2018}$$ is between two consecutive perfect squares?

#### Idea

sequence of unit digits of the $T_i$

$$\displaystyle \{1,2,3,7,3,7,3,7,3,7,3,7,3,7,\text{...}\}$$

so it looks like $T_{2018}$ ends in a $7$

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