There is something very special about the number 10311.
Recall that a (natural) number is perfect if it is the sum of its proper divisors. (A proper divisor of a number n is a divisor of n other than n itself.) For example, 6 = 3 + 2 + 1 is perfect. In general, if d1, ..., dk are the proper divisors of n, then n is perfect if
n = d1 + ... + dk (Equation 1)10311 is not perfect: its proper divisors are 1, 3, 7, 21, 491, 1473, 3437, which sum to 5433. (Since 5433 < 10311, 10311 is said to be "deficient".) However, consider the reversal of 10311, that is, 11301, and the reversals of these proper divisors: 1, 3, 7, 12, 194, 3741, 7343. Adding the proper divisor reversals 1 + 3 + 7 + 12 + 194 + 3741 + 7343 = 11301 yields the reversal of 10311!
Therefore, for the number 10311, equation 1 holds if each term is replaced by its reversal.
The reversal of a number can be thought of as the "mirror image" of that number. This motivates the term "picture-perfect number" to describe a number such as 10311. More precisely, n is picture-perfect if the reversal of n is the sum of the reversals of the proper divisors of n. Hence, if n is written on one side of an equal sign and its proper divisors on the other, then the resulting equation read backwards is valid.
In my paper On a Generalization of Perfect Numbers, I introduced "f-perfect numbers", that is, perfect numbers relative to an arithmetical function f. The picture perfect numbers are f-perfect numbers for f(n) = Reversal(n). This perfect number set is the first one I have considered the definition of which depends on a particular base (base 10 here).
Here is Mathematica code to find the picture-perfect numbers not exceeding 108:
f[n_]:= FromDigits[Reverse[IntegerDigits[n]]];
Do[If[f[n]==Apply[Plus,Map[f,Drop[Divisors[n],-1]]],Print[n]],{n,2,10^8}]The picture perfect numbers are extremely rare. In fact, 6 and 10311 are the only such numbers not exceeding 107. Considering that 6 and its proper divisors are one-digit numbers, the reversals of which are themselves, 6 can be seen as trivially picture-perfect. Hence, the only non-trivial picture-perfect number not exceeding 107 is 10311.
These observations bring us to our challenge: can you find another picture-perfect number? Or is 10311 the only (non-trivial) one? Are there only finitely many picture-perfect numbers? If not, can you find an expression generating them?
Update, 11 April 2002:
After seven hours of running Mathematica on my machine (600 MHz Pentium III), I was rewarded with the third picture-perfect number! It is the eight-digit 21661371. There are no other picture-perfect numbers not exceeding 25 million.
The proper divisors of 21661371 are 1, 3, 9, 27, 443, 1329, 1811, 3987, 5433, 11961, 16299, 48897, 802273, 2406819, 7220457. Reversing these divisors then adding, we get the reversal of 21661371:
1 + 3 + 9 + 72 + 344 + 9231 + 1181 + 7893 + 3345 + 16911 + 99261 + 79884 + 372208 + 9186042 + 7540227 = 17316612.
Mark Ganson has conjectured that all picture-perfect numbers are divisible by 3. So far, his conjecture is holding water. Let's see what insights the discovery of the fourth picture-perfect number, if it exists, will bring.
Update, 12 April 2002:
Daniel Dockery writes:
I thought you might like to know that I have extended the search through 255 million and have not yet found any others than those you've mentioned. I will continue the search when I have the CPU cycles to spare, but at the present it looks like any other picture-perfect numbers will be greater than 255000000.
Thanks for the interesting challenge.
Update, 18 April 2002:
Today, I have had the pleasure of discovering the smallest "picture-amicable" pair (i.e., Reversal(n)-amicable pair; see the above link to my paper for the definition): 2320000, 34049. Here is Mathematica code to generate picture-amicable pairs:
f[x_] := FromDigits[Reverse[IntegerDigits[x]]];
d[x_] := Apply[ Plus, Map[ f, Drop[Divisors[ x], -1] ] ];
i = 2;
Do[If[i==100000, i=0;Print["Checked all n up to " <> ToString[n]]];i++;
a = d[n];
b = f[a];
c = d[b];
If[Mod[a, 10] != 0 && n != b && c == f[n], Print[{n, b}]], {n, 2, 10^8}]Join the fascinating search for more of these elusive pairs!
Update, 22 April 2002:
There is now a coordinated effort to find more picture-perfect numbers. The SUPER (Systematic Undertaking to find Picture-Perfect Numbers) team has a message board to announce findings and discuss ideas:
We invite you to join us in this exciting search!
Update, 26 April 2002:
Congratulations to Daniel Dockery for finding the fourth picture-perfect number. It's the ten-digit
1460501511.
This can be verified by reading the following equation backwards:
1460501511 = 1 + 3 + 7 + 21 + 101 + 303 + 707 + 2121 + 688591 + 2065773 + 4820137 + 14460411 + 69547691 + 208643073 + 486833837.
The new number is also divisible by 3, further strengthening Ganson's conjecture.
Update, 30 May 2002:
Here is the first draft of my paper summarizing the current state of the art on picture-perfect numbers.
Joseph L. Pe
iDEN System Engineering Tools and Statistics
Motorola Center
Schaumburg, IL
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Document created on 9 April 2002 by J. L. Pe. Last updated on 1 November 2002.