# Want Some Really Big Numbers? Here Are the Zoogol and the Zoogolplex.

Portmanteaus of the words “zillion” and “googol” / “googleplex,” these two new number-names are offered for the use of anyone who feels a need for even more named numbers which are ridiculously larger than the number of particles known to exist. The vast majority of those particles are neutrinos, by the way, but fewer than a googol (let alone a zoogle) neutrinos are estimated to exist in the entire observable universe.

To write a zoogle out the long way, simply write a “1,” then follow it with a “mere” one million zeroes. To my knowledge, this has never been done, but it certainly is possible. To write out a zoogleplex the long way, you’d need to follow the “1” with a zoogol zeroes, but this is not possible, due to a lack of enough matter or space, in the entire universe, for such a task.

Pop quiz: which is larger: a googleplex, or a zoogle? Scroll down to find the answer, whenever you are ready.

(keep scrolling….)

A googolplex is vastly larger than a zoogol. However, all other numbers mentioned earlier in this post are dwarfed by a zoogolplex. However, even a zoogolplex is less than 1% of 1% of 1% of . . . 1% of infinity, no matter how long one makes the “of 1%” chain.

# New Largest Known Prime Number Revealed By Computer — After Four-Month Delay

I find it hilarious that the computer which made this discovery actually kept it a secret for four whole months.

From that article: “[Curtis] Cooper’s computer actually found the prime on 17 September 2015, but a bug meant the software failed to send an email alert reporting the discovery, meaning it went unnoticed until some routine maintenance a few months later.”

# A Set of Conjectures: Sequels to Fermat’s Last Theorem?

This story began yesterday, with this blog-post: https://robertlovespi.wordpress.com/2014/12/10/pythagorean-and-fermatian-triples-and-quadruples/ — but it hasn’t ended there. When discussing this with my wife (who, like myself, is also a teacher of mathematics) while writing that post, she speculated that more interesting things might happen — such as a “no solutions” situation, as is the case with Fermat’s Last Theorem — with a search for a Fermatian quadruple, if the exponent used were larger than three, the exponent I checked yesterday.

Tonight, therefore, I modified the program I used for the last post on this subject. Instead of searching for whole-number solutions to an + bn + cn = dwith n = 3, as I did yesterday, I looked for solutions to a4 + b4 + c4 = d4. As I did yesterday, I started with a search of all possibilities with numbers from 1 to 10, and was unsurprised when that quick, preliminary search yielded no solutions. I then ran the program again, but used it to search all possibilities using numbers for a, b, c, and d from 1 to 100. This took a while, for, with loops nested four deep, my computer had to check 1004 = 100 000 000 possibilities. The results are — tentatively — exciting, for this search, indeed, yielded no solutions, which is reminiscent of Fermat’s Last Theorem:

We are now wondering if Fermat’s Last Theorem can be generalized indefinitely. Andrew J. Wiles proved that an + bn = cn has no solutions if n > 2. I’ve now written a program, and checked, and know that an + bn + cn = dn  has no solutions, with n = 4, for values of a, b, c, and d up to 100.

Could it be that an + bn + cn = dhas no solutions for any value of n > 3 — like Fermat’s Last Theorem, but with one more term added, and the exponent simply bumped up one place? If that is true, then, might it also be true that an + bn + cn + dn = ehas no solutions if n > 4? Might it be possible to extend this idea indefinitely, so that, with an equation containing k terms added together, to equal a single term, there are no solutions if n > k?

I know this much, at this point:

1. I can either find a counterexample, to disprove one of these conjectured “sequels” to Fermat’s Last Theorem, if that counterexample is of reasonable size, provided a “smallish” counterexample actually exists, or
2. With the assistance of friends of mine whose ability with mathematics, and computer programming, exceeds my own, we can extend this search for counterexamples to much higher limits, or
3. This set of conjectures is, in fact, true, in which case we will find no counterexamples — and, if that is the case, I’m going to need to find some major-league help for this problem, for, well, if true, it’s going to be one monster of a job to prove it is true, especially in the general, unlimited form.
4. Finally, I know that the prospect of playing any role, whatsoever, in extending Fermat’s Last Theorem to new levels is tremendously exciting.

I’m looking forward to seeing where this goes.

[Update: I’d like to thank my friend Andrew for finding the answer to this puzzle for me. Counterexamples have indeed been found for the four-term and five-term cases, one of which is 26824404 + 153656394 + 187967604 = 206156734. For six terms or more, this remains an unsolved problem. For more information related to this, please visit https://en.wikipedia.org/wiki/Euler’s_sum_of_powers_conjecture.]

# Pythagorean and Fermatian Triples and Quadruples

Pythagorean triples are familiar to almost everyone who has studied mathematics:  whole numbers which serve as solutions to the Pythagorean Theorem, a² + b² = c². Examples include 3, 4, 5; as well as 5, 12, 13; and 8, 15, 17; and 7, 24, 25. It has been proven that there are infinitely many Pythagorean triples.

Fermatian triples, on the other hand, don’t exist, which humanity finally found out, definitively, when Andrew J. Wiles finally proved Fermat’s Last Theorem, in the mid-1990s. If they did exist, they would satisfy an + bn = cn, with n > 2, and all numbers involved being whole numbers. This “only” took over 300 years to prove, and will forever stand as one of the greatest achievements in number theory.

This morning, while driving to work (one must think about something while driving, right?), I started trying to find Pythagorean quadruples:  sets of four whole numbers which satisfy a² + b² + c² = d², which can be pictured as pairs of solutions to the Pythagorean Theorem, for right triangles in which the hypotenuse of one triangle is a leg of the next, and the triangles exist in perpendicular planes. It didn’t take me long to figure out that 3, 4, 12, 13 is a Pythagorean quadruple, based on this mental image:

The next logical step was to wonder . . . are there Fermatian quadruples? Those would be, of course, whole-number solutions to an + bn + cn = dn, with n > 2. However, I had to teach all day, and did not have the time, until after work, to indulge my curiosity on this subject.

Once the workday was over, I contemplated looking over lists of perfect cubes (since three as the exponent is the logical place to start looking), seeking three of them that would sum to a fourth, and quickly decided that was not the approach I wanted to use . . . because it sounded ridiculously tedious. Mathematics is supposed to be fun, after all, not an exercise in boredom. I therefore resolved to use a different approach, and wrote a short program, in BASIC, to check all sets of four numbers between one, and any number I chose, to seek exponent-three Fermatian quadruples. For the first trial run of this program, I considered checking all numbers between 1 and 100, but, since the program involves quadruple-nested loops, I decided I did not want to wait for my computer to check 100^4 = 100 million different combinations, so I made my first check with much more modest search parameters: only the numbers from one to ten. To my surprise, this search actually revealed two Fermatian quadruples.

The two Fermatian quadruples this search revealed are 1, 6, 8, 9 (since 1 + 216 + 512 = 729); and 3, 4, 5, 6 (since 27 + 64 + 125 = 216). With a more extensive search, I could easily find more, and I suspect there are infinitely many of them, as is the case with Pythagorean triples . . . but this is enough recreational mathematics, I think, for one day.

[Later edit — to see what happened the next day, with this idea, just check this post: https://robertlovespi.wordpress.com/2014/12/11/a-set-of-conjectures-sequels-to-fermats-last-theorem/.]

# A Graphical Survey of Prime, Perfect, Deficient, and Abundant Numbers From Two to Thirty

In this graph, each number on the x-axis (from 2 to 30) is plotted against the sum of all its factors (including one, but excluding the number itself) on the y-axis. Numbers on the blue line y = 1 have no factors other than one and themselves, and are therefore prime numbers. Numbers on the green line y = x are equal to the sum of their factors (including one, but excluding themselves), and are therefore perfect numbers. Perfect numbers are much rarer than prime numbers in the entire set of natural numbers, as well as in this small sample.

If a number’s factor-sum, examined in this manner, is smaller than the number itself, such a number is called a “deficient number.” This applies to all numbers with points below the green line. Numbers which have points on the blue line are deficient numbers, as well as being prime numbers – and this is true for all prime numbers, no matter how large. The numbers represented by points between the green and blue lines are, therefore, both deficient and composite, and can also be called “non-prime deficient numbers.”

A few numbers on this graph, called “abundant numbers,” are represented by points above the green line, because their factor-sum is greater than the number itself. There are only five abundant numbers in this sample: 12, 18, 20, 24, and 30. As an example of how a number is determined to be abundant, consider the factors of 30: 1+2+3+5+6+10+15 = 42, which is, of course, greater than 30.

Of the 29 numbers examined in this sample, here is how they break down by category:

• Abundant numbers: 5 (~17.2% of the total)
• Perfect numbers: 2 (~6.9% of the total)
• Non-prime deficient numbers: 12 (~41.4% of the total)
• Prime numbers: 10 (~34.4% of the total)

These percentages only add up to 99.9%, due simply to rounding. Also, the total number of deficient numbers in this sample (both prime and composite) is 22, which is ~75.9% of the total sample of 29 numbers.

So what happens if this survey is extended far beyond the number 30, to analyze much larger (and therefore more meaningful) samples? Well, for one thing, the information on the graph above would quickly become too small to read, but that is only of trivial importance. More significantly, what would happen to the various percentages, for each category, given above? First, both prime and perfect numbers become more difficult to find, as larger and larger numbers are examined – so the percentages for these categories would shrink dramatically, especially the one for perfect numbers. With smaller percentages of prime and perfect numbers in much larger samples, the sum of the percentages for the other two categories (abundant and non-prime deficient numbers) would, of necessity, grow larger. That has to be true for this sum – but that says nothing about what would happen to its two individual components. My guess is that abundant numbers would become more common in larger samples . . . but since I have not yet examined the data, I’m only calling this a guess, not even a conjecture. As for what would happen to the percentage of non-prime deficient numbers when larger samples are analyzed, I don’t even (yet) have a guess.

# A Number Theory Puzzle Involving Primes, Perfect Numbers, and “Paraperfect” Numbers

There is something about the definitions of prime and perfect numbers that always struck me as rather odd. Prime numbers are those which no factors other than themselves, and one. Perfect numbers, on the other hand, equal the sum of all of their factors, excluding themselves, but including one. The first two examples of perfect numbers are 6 (which equals 1 + 2 + 3) and 28 (which equals 1 + 2 + 4 + 7 + 14). Perfect numbers are far more rare than primes.

The thing I find annoying is the exclusion of one, as a factor, from one of these definitions, but not the other. I therefore decided to give a name to a new type of number:  one which equals the sum of its factors, excluding itself AND the number one. The first name I thought of, “exceptional numbers,” turns out to have already been taken, so I thought of another, and called these numbers “paraperfect numbers” instead.

Having done that, it was time to start searching for them. I have a reasonably fast mental calculator, but it didn’t take long to figure out that I wasn’t up to this task, so I wrote this program to search for paraperfect numbers:

It’s written in BASIC, an archaic computer language I learned in high school, and, as you can see, I am a horribly sloppy programmer. A better programmer would have written a program for the same purpose, but with only about half this length. Be that as it may, though, the program does work. I’ve had it running for a few minutes now.

It’s gotten past 22,000 — and has found no paraperfect numbers at all. This is not what I expected . . . and now I am wondering if any exist. Right now, of course, “no paraperfect numbers exist” is a mere conjecture. If I can prove it, it will be a theorem. However, I don’t know nearly enough about number theory to write such a proof.

I could use some help. If anyone does find a paraperfect number, please leave a comment on this post identifying your find. If anyone can prove — or simply explain to me — why there are no paraperfect numbers, if that is the case, please let me know that as well.

The program is still running, and has now passed 25,000 without a paraperfect-number find. I guess I’ll leave it running for a while. Any help with this puzzle would be much appreciated.

[Later:  see the comments for the rest of the story on these elusive numbers.]