Cubes of Perfection

Playing with integers can lead to all sorts of little surprises.

A whole number that is equal to the sum of all its possible divisors—including 1 but not the number itself—is known as a perfect number. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 equals 6.

Six is the smallest perfect number. The next is 28. Its proper divisors are 1, 2, 4, 7, and 14, and the sum of those divisors is 28.

Incidentally, if the sum works out to be less than the number itself, the number is said to be defective (or deficient). If the sum is greater, the number is said to be abundant. There are far more defective and abundant numbers than perfect numbers. Do abundant numbers actually outnumber defective numbers? I'm not sure.

Steven Kahan, a mathematics instructor at Queens College in Flushing, N.Y., has a long-standing interest in number theory and recreational mathematics. "I often play with number patterns," he says.

In the course of preparing a unit on number theory for one of his classes, he noticed a striking pattern involving the perfect number 28: 28 = 1³ + 3³. He tried the next perfect number: 496 = 1³ + 3³ + 5³ + 7³.

More than 2,000 years ago, the Greek geometer Euclid of Alexandria (325-265 B.C.) proved that if 2^p - 1 is a prime number, then 2^{p - 1}(2^p - 1) is an even perfect number. Primes of the form 2^p - 1 are now known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime.

Leonhard Euler (1707- 1783) proved the converse of Euclid's theorem: All even perfect numbers must have the form specified by Euclid's formula. Hence, every Mersenne prime automatically leads to a new perfect number. There are, at present, 48 known Mersenne primes.

It turns out that a given perfect number 2^{p - 1}(2^p - 1) greater than 6 is expressible as the sum of the cubes of the first n consecutive odd integers, where n = 2^{(p - 1)/2}. For example, if p = 7, then n = 8, and the perfect number 8,128 equals the sum of the cubes of the first eight odd integers: 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³. Kahan provides a short proof of this venerable theorem in the April 1998 Mathematics Magazine.

This means that the largest known perfect number, which has 34,850,340 digits, is the sum of the cubes of the first 2^28942580 consecutive odd integers.

Perfect numbers also show other curious patterns. Add the digits of any perfect number greater than 6, then add the digits of the sum together, and so on, until only one digit remains. That final digit is always 1.

28: 2 + 8 = 10; 1 + 0 = 1

496: 4 + 9 + 6 = 19; 1 + 9 = 10; 1 + 0 = 1

8,128: 8 + 1 + 2 + 8 = 19; 1 + 9 = 10; 1 + 0 = 1

Here's another remarkable relationship. The sum of the inverses of the divisors of a perfect number (leaving out 1 but including the number itself) is also 1.

6: 1/2 + 1/3 + 1/6 = 1

28: 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 1

496: 1/2 + 1/4 + 1/8 + 1/16 + 1/31 + 1/62 + 1/124 + 1/248 + 1/496 = 1

Interesting digit patterns and numerical relationships also arise when Mersenne primes and perfect numbers are written out in binary form. Mersenne primes, for example, consist of unbroken strings of consecutive 1s—57,885,161 of them in the case of the current record holder.

Here are the first four perfect numbers: 110, 11100, 111110000, 1111111000000. See a pattern?

Happy hunting in perfect territory!

Originally posted May 18, 1998

References:

Gullberg, J. 1997. Mathematics: From the Birth of Numbers. W.W. Norton.

Kahan, S. 1998. Perfectly odd cubes. Mathematics Magazine 71(April):131.

Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. Holt.

An introduction to Mersenne primes and perfect numbers can be found at http://www.utm.edu/research/primes/mersenne.shtml.

Fermat's Natural Spirals

A typical daisy is a star-like flower. It features a fringe of white or colored petals and a central disk of tubular florets. Each floret is itself a tiny flower.

This daisy has 21 white petals and a yellow central disk of tubular florets. Photo by Kenneth Peterson.

The tightly packed florets at a daisy's center have an intriguing arrangement. The florets get larger at greater distances from the center. And there are hints of clockwise and anticlockwise spirals in the pattern.

A close-up view of a daisy's disk of florets reveals intriguing spiral patterns. Photo by Kenneth Peterson.

One way to model such a pattern is to start with a curve called Fermat's spiral. This curve is also known as a parabolic spiral. It's given by the polar equation

r = k a^1/2

where r is the distance from the origin, k is a constant that determines how tightly wound the spiral is, and a is the polar angle.

This type of spiral has the property of enclosing equal areas with every turn.

Fermat's spiral.

By placing points (disks or polygons) centered at regular angular intervals along such a spiral, you can create a variety of intriguing patterns—depending on the angle you choose to use. Using the angle 222.49 degrees (a value related to the golden ratio, 1.618034. . . ), you get a pattern with an even packing of polygons (or disks). It closely resembles a daisy's florets.

By placing points at regular angular intervals along Fermat's spiral, you get a pattern resembling that of a daisy's central florets. Courtesy of Robert Krawczyk.

By choosing other angles, you get intriguing variants. Each choice gives a different pattern of secondary spirals, some winding clockwise and others anticlockwise, which form an interlocking system. Robert Dixon explores some of these possibilities in his book Mathographics.

Using larger numbers of points and smaller angles produces patterns with a variety of secondary spirals and, often, with radial lines that become evident toward the edges. Michael Naylor has investigated a variety of such patterns (see "Golden Blossoms, Pi Flowers").

Robert J. Krawczyk has taken the generation of these patterns another step further, creating striking images of eerie ripple patterns. He calls these circular designs "Fermat's spiral mandalas."

Krawczyk starts by combining several spirals to create one complex pattern.

Four different individual spirals (top) can be combined in various ways to create new, complex patterns (bottom). Courtesy of Robert Krawczyk.

By placing points at fixed angular intervals along these curves, he gets very elaborate patterns that show a variety of features.

An example of a Fermat's spiral mandala. Courtesy of Robert Krawczyk.

To finish his images, Krawczyk enhances the texture and gives them a coppery glow. Examples of such mandalas can be seen at his website.

Courtesy of Robert Krawczyk.

A mandala's complex circular design, with its symmetrical and radial balance, is intended to draw the eye to its center, Krawczyk says. Fermat's spiral, in particular, "is a natural basis for this inward draw."

For those who prefer a more natural look, all it takes is a close view of a daisy's central disk.

Photo by Kenneth Peterson.

Originally posted September 5, 2005.

References:

Dixon, R. 1991. Mathographics. Dover.

Krawczyk, R.J. 2005. Fermat's spiral mandalas.

Naylor, M. 2002. Golden, √2, and π flowers: A spiral story. Mathematics Magazine 75(June):163-172.

Designer Decimals

Calculate 100/89. You get the decimal expansion 1.1235955056 . . .

Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .

This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).

Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.

Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.

James Smoak discovered this curious phenomenon, and he and Thomas J. Osler went on to prove that this class of fractions always produces decimal expansions containing terms of the Fibonacci sequence. They described it as "a magic trick from Fibonacci."

A little later, Marjorie Bicknell-Johnson found a formula, or "generalized mathematical magician," that identifies fractions whose decimal representations include successive values belonging to a variety of other sequences. She called them designer decimals.

Smoak (with O-Yeat Chan) then continued his adventures in the realm of designer decimals, as reported in the November 2006 College Mathematics Journal.

Consider, for example, the fraction 10000/9801. It has the decimal expansion 1.0203040506 . . . , suggesting the existence of a new class of fractions with curious properties.

Smoak and Chan ask: Do all the integers from 1 to 99 occur in the sequence? Given that the decimal expansion must repeat, what is the length and nature of the repeating part?

The key, Smoak and Chan say, is to note that 9801 = 99². So 10000/9801 = (100/99)² = (1.0101010101...)².

Then, it's possible to show that the repeating part is 0203 . . . 97990001.

In general, fractions of the form [10ⁿ/(10ⁿ – 1)]^k yield the sequence of integers in their decimal expansions.

It's amazing what can lie hidden in simple fractions!

Originally posted November 6, 2006

References:

Bicknell-Johnson, M. 2004. A generalized magic trick from Fibonacci: Designer decimals. College Mathematics Journal 35(March):125-126.

Chan, O-Y., and J. Smoak. 2006. More designer decimals: The integers and their geometric extensions. College Mathematics Journal 37(November):355-363.

Smoak, J., and T.J. Osler. 2003. A magic trick from Fibonacci. College Mathematics Journal 34(January):58-60.

Spreading Rumors

Three people take longer to share their gossip than four people!

This curious result arises out of the following situation: A group of friends love sharing their gossip. Each gossiper initially knows something that no one else in the group knows. Each day, some of the gossipers phone each other and exchange all the news they have collected so far. On a given day, however, each gossiper can participate in only one phone call, and no conference calls are allowed.

What is the minimum number of days required for all gossipers in the group to learn all the news?

If the group consists of only one member, the minimum number of days is zero; for two people, only one call is required to share two pieces of gossip. A group of three people needs three days to get everyone up to speed, and a group of four people takes only two days.

Here's the table that C. Kenneth Fan, Bjorn Poonen, and George Poonen produced to illustrate these results:

Number of gossipers	1	2	3	4	5	6	7	8
Minimum number of days	0	1	3	2	4	3	4	3

The minimum number of days doesn't increase steadily as the number of participants grows. From the pattern, you can see that it would probably be helpful to consider odd and even numbers of participants separately.

Suppose a group has an even number of gossipers. On the first day, the group can be divided in half. Each of the two subgroups will learn all the gossip within the subgroup in a certain number of days. Gossipers in one subgroup can then be paired with gossipers in the other subgroup so that, in another day, all the gossipers know all the gossip.

It turns out that if the number of gossipers, n, is 2^D, then the minimum number of days required to spread the rumors to everyone in the group is D. Thus, four (2²) participants would take two days; and eight (2³) gossipers would require three days.

Now, consider a group of seven gossipers: Alice, Bob, Carol, Daniel, Eric, Fred, and Greta. One way to spread the rumors is to divide the group into two subgroups, one consisting of four members (Alice, Bob, Carol, and Daniel) and the other of three members (Eric, Fred, and Greta). On the first day, Alice phones Eric, Bob phones Fred, and Carol phones Greta, while Daniel takes a break. At this point, Alice, Bob, Carol, and Daniel collectively have all the information available, and it takes them two days to share it among themselves. Then, Alice calls back Eric, Bob calls Fred, and Carol calls Greta to complete the information transfer. Total time: four days.

Illustration of how rumors can spread in four days among seven gossipers. The numbers indicate the day on which the connected couple communicates.

Four days is the best that a group of seven can do no matter what the strategy.

In an article in Mathematics Magazine, Fan, Poonen, and Poonen prove that, in general, the minimum number of days for an even number of participants to share information is given by the exponent of the power of two exactly representing the number of gossipers or by the exponent of the next higher power of two. Thus, four (2²) gossipers would need two days, six would need three days (the next higher power of two is 2³), eight would need three days, and 10 would need four days (because 10 is greater than eight but smaller than 16, or 2⁴).

For an odd number of participants, just add one day to the number of days required by the next higher even number of gossipers.

Fan, Poonen, and Poonen readily concede that their mathematical model of rumor mongering may be somewhat unrealistic. "Admittedly, one day is a bit much for the time it takes for even the most loquacious of rumor mongers to communicate with a friend!" they commented. "Perhaps one hour or even one minute would have been more realistic."

Interestingly, the problem can also be interpreted in terms of communication between computers operating in parallel. "The result can be useful, for instance, when multiple copies of a database are distributed to several locations and replication is needed to keep the databases synchronized," the mathematicians said. "In fact, this is the situation in which our problem originally arose."

References:

Fan, C.K., B. Poonen, and G. Poonen. 1997. How to spread rumors fast. Mathematics Magazine 70(February):40-42.

Peterson, I. 2002. Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.

Originally posted March 17, 1997