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

graphical survey of prime, perfect, deficient, and abundant numbers from 2 to 30

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.

All-Seeing Eye

all-seeing eye

Black and White Tessellation Using Semicircles

Image

tess curves

All of the curves used in this tessellation are semicircles. The tessellation itself is a modification of the regular tessellation of the plane using equilateral triangles.

Forgiveness: Not a Virtue, But a Dangerous Practice

Over the millennia, religion has done much harm, in myriad ways. Of the major world religions, the one that places the greatest emphasis on forgiveness is, to my knowledge, Christianity. This was an error in reasoning made many centuries ago, and it is impossible to calculate the amount of harm this doctrine has caused . . . but the number of people harmed by this terrible idea certainly numbers in the millions.

Consider one of the most oft-quoted passages from the New Testament concerning this topic, from Matthew 18:21-22 (NASB):  “Then Peter came and said to Him, ‘Lord, how often shall my brother sin against me and I forgive him? Up to seven times?’ Jesus said to him, ‘I do not say to you, up to seven times, but up to seventy times seven.’” Seventy times seven is, of course, 490, but it is rare to find anyone who takes that number literally. It is much more common to encounter the explanation that large numbers were viewed differently in the ancient world, and “seventy times seven” was simply a way for Jesus to say, in a way Peter would understand, “an indefinitely large number.”

Now, consider what we know about the modern world. At least one-third of women are raped during their lifetimes. Serial killers often murder dozens of people before getting caught. Powerful people, in positions of public trust and great responsibility, betray that trust for their own selfish reasons. This list could be much longer, but I trust the point has been made:  you live in a world with many others in it who are not nice people . . . and many of them have no intention of changing.

Consider this:  a newly-married woman discovers her husband is betraying her in one of the worst possible ways, by sexually molesting children who live in nearby homes. She decides to leave him, and contacts her (devoutly religious) family, asking for help – only to be told that marriage is a sacred covenant, divorce is a sin, and the evil deeds of others are, according to the Bible, supposed to be forgiven. “Pray for him,” she is told — but the real support she is asking for is not given. She tries to forgive him. She stays in the marriage for many more years. The unsurprising result? Dozens more children are abused by the man over the following decades, with far-reaching, horrible consequences.

That last example was not hypothetical. The woman, and her family, are people I know.

There are people – many of them – who simply do not deserve to be forgiven for the crimes they commit. They are dangerous, and will remain so, until and unless they are stopped. Some stop only when they die — and those deaths, I do not mourn. Others are caught, tried, convicted, and imprisoned. However, those people are, too often, released while still dangerous, due to another nonsensical idea (that of having paid one’s “debt to society”), or simply because prisons are overcrowded with many people who only committed non-violent illegal acts. Both problems are easy to solve, however. First, we should stop locking up non-violent offenders – that’s the obvious part of the solution. The other part is more difficult, for it would require major legislative changes:  the abolition of specific, time-limited sentences for violent criminals.  Why lock up, say, a murderer or rapist for ten years, and then let them go, more dangerous than ever? It would make more sense to leave such people – anyone who is clearly dangerous to the rest of us – locked up for life, or at least until they have become so weakened by illness or advancing age that they are no longer capable of harming other people.

What about lesser offenses? What if, for example, you catch someone you know in a harmful, deliberate, and malicious lie? Should you forgive them? My answer is often a flat “no” – at least, not until the person has regained the trust they have damaged or destroyed, and sometimes that simply is not possible. (Who decides when trust is restored? The person who was lied to, of course.) Forgive a pathological liar, and what you are really doing is inviting them to lie to you again. A far better thing to do would be to warn others not to trust the liar, and explain exactly why that is the case.

Some who wish to cling to their religious beliefs, even when those very beliefs cause obvious problems, have devised a way to try to get around the problem that forgiving those who harm you, or your loved ones, invites further harm. You are likely to have heard it, or something like it:  “I forgive them, but I will not forget what they have done, for they may well do it again, and I must be on my guard.” Such a statement is an improvement over total, unconditional forgiveness, but it is not without problems. First, if one is constantly vigilant for a repeat offense, has forgiveness really taken place? Not by the Biblical standard of divine forgiveness of the evil deeds of people, it hasn’t, as Hebrews 8:12 (NIV) makes clear: “For I will forgive their wickedness and will remember their sins no more.” Many other verses tie the acts of forgiving and forgetting together. Separating the two, as many people do now, is an improvement, to be certain . . . but it is in no sense an idea rooted in either the Bible, or in traditional Christian doctrine. It is, instead, a modern concession to reason and common sense.

What about really small things?  Accidents, honest mistakes . . . that sort of thing?  Is there a problem with forgiveness in those sorts of situations?  No, there isn’t . . . but there also would have been no problem with not getting angry at a person for such a “crime” in the first place. As a good rule of thumb, if it made perfect, rational sense to get angry at someone because they did something truly terrible, then it does not make sense to forgive them for it ten minutes later, nor the next day . . . perhaps not even until they die, because at that point, the chances of them repeating the offense drops to zero. In other words, it isn’t yet time to forgive a person who still poses a danger. This is simple logic.

Monsters in human form, like everything and everyone else, are part of the physical universe. If one or more of them does something terrible to you, or to someone you care for, it makes sense to take steps to prevent their repeating such an act. It does not make sense to forgive them for it. To do so is the equivalent of telling the universe that you want you or your loved ones to have to endure further suffering. That is not a logical way to live one’s life. “Forgiveness is a virtue” is a pernicious idea – one we should, as a species, leave in the past, if we want to make progress in the future.