# Puzzling Palindromic Year with Pleasing Palindromic Expression.. Please find them

No computers, calculators or searches.

Given:

The number is a Palindromic year. $$PQ$$ is a concatenated number.

$$U , V$$ are two distinct positive integers

Expressed as

$$PQ ^ U + PQ ^ V + PQ ^ V + PQ ^ U$$

Find at least two other Palindromic Properties associated with this number !

• Answer should also include at least two other palindromic properties associated..it could be another expression, visual symmetry etc.. – Uvc May 17 '19 at 11:41

One possible solution

$$2112$$

Reasoning

Without loss of generality, suppose $$U < V$$, then we can simplify the expression to $$2(PQ^U)(PQ^{V-U}+1)$$ Given that we are looking for a "year" and that this is also even, it is likely that the year begins with $$2$$.
Then, we can look at the simplest case where $$U=V-U=1$$ and the expression becomes $$2(PQ)(PQ-1)$$ If we start with $$2002$$, we see that this factorises as $$2 \times 7 \times 11 \times 13$$ and can determine relatively quickly that this cannot match the above expression.
Continuing on with $$2112$$, we see that this factorises as $$2^6 \times 3 \times 11 = 2 \times 2^5 \times 3 \times 11 = 2 \times 32 \times 33$$ which is conveniently in the form above.
Hence we have $$PQ = 32 \,\,\,\,,\,\,\,\, U=1\,\,\,\,,\,\,\,\, V=2$$

Two other palindromic properties:

1. If we write $$2112$$ on a digital display and rotate $$180^o$$, the result is again $$2112$$.
2. The number $$2112$$ can be expressed as a palindromic product, that is $$2112 = 2 \times 33 \times 32 = (2)(33)(32)$$

• Please see added comment. – Uvc May 17 '19 at 11:42
• Awesome!....... – Uvc May 20 '19 at 8:56