Guide Cool Numbers

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All submissions for this problem are available. Consider some examples. Note that here we select two digits: 1 and 5. Now consider some positive integer N of n digits. Input The first line of the input file contains an integer T , the number of test cases. Example Input: 4 2 Output: 2 3 Explanation Case 1. CodeChef is a non-commercial competitive programming community.

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If there is a score for the problem, this will be displayed in parenthesis next to the checkmark. Time Limit Exceeded Your program was compiled successfully, but it didn't stop before time limit. Try optimizing your approach. Wrong Answer Your program compiled and ran succesfully but the output did not match the expected output. Runtime Error Your code compiled and ran but encountered an error. The most common reasons are using too much memory or dividing by zero.

Finally, I disqualified number that were highly non-canonical in construction. That said, I felt free to ignore any of these disqualifications when I felt like it. I hope you enjoy the following list, and I welcome feedback.

Cool Numbers

Honorable Mentions. If it turns out to be the largest Fermat prime, it might earn itself a place on the list, by virtue of thus also being the largest odd value of n for which an n -gon is constructible using only a rule and compass. Specifically, the largest known collection of sociable numbers has cardinality 28, and though this might seem a silly feat in and of itself, the fact that sociable numbers and perfect numbers are so closely related may reveal something slightly more profound about 28 than it just being perfect.

Only slightly less canonical is that it gives the limiting ratio of subsequent Fibonacci numbers. Also, this number plays an important role in the hstory of algebraic number theory.

Multiplication/Division

The field it generates is the first known example of a field in which unique factorization fails. Trying to come to grips with this fact led to the invention of ideal theory, class nubers, etc. There are 17 positive integers smaller than 78, not yet known to be non-Sierpinski. Though the choice of 2 here for the exponent is somewhat non-canonical i.

The one thing going for it is that it seems to be highly canonical, representing the limiting ratio of distance between bifurcation intervals for a fairly large class of one-dimensional maps. In other words, all maps that fall in to this category will bifurcate at the same rate, giving us a glimpse of order in the realm of chaotic systems. The question here is canonicality.

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Yet 2 has some remarkable features even ignoring aspects relating to its primality. Finally, if nothing else, it is certainly the first prime, and could at least be included for being the first representative of such an amazing class of numbers. The above integer is the size of the monster group , the largest of the sporadic groups. This gives it a relatively high degree of canonicality. Since there is , however, there must be something fairly special about the largest possible one. This seems to represent something fundamental about the harmonic series, and thus of the integers themselves.