(Note: I got the idea for this essay from a spoof of inductive reasoning that I read in a book many years ago, although this particular development of the idea is my own. I regret that I cannot remember either the title or the author of the book and so cannot give appropriate credit. If anyone reading this happens to remember seeing anything like it and can remember where, I would appreciate hearing from them.)
Mathematicians traditionally have classified all whole numbers into two kinds, prime and composite. According to this traditional classification, prime numbers are divisible only by themselves and unity, while composite numbers have other divisors. Unity -- "one" -- is considered neither prime nor composite, for reasons not relevant to this essay.
Since all even numbers are divisible by 2, obviously no even number other than 2 itself can be prime. By definition, therefore, all even numbers except 2 are composite.
In that case, we should expect all odd numbers to be prime. We have conducted an investigation and found this to be true, despite the claims of many skeptics. We are presenting the evidence here. We encourage all people who are honestly open-minded to examine the facts themselves and make their own decisions. We are confident that anyone who sincerely desires to know the truth will see it in these facts.
We begin by examining all the odd numbers below 100.
By definition, unity is a special case, neither prime nor composite.
3 x 3
Its square root is prime. That is a type of primality.
3 x 5
Its factors are twin primes. That is like having a prime square root.
At this point, it is clear that prime numbers may be divided into two classes, defined as follows. A strong prime has no factor besides itself and unity. A weak prime is the square of a strong prime or the product of twin primes.
Therefore, 15 is a confirmation of the theory.
7 x 3: Anomaly
5 x 5: Square of a prime.
3 x 3 x 3: Perfect cube, and its cube root is prime. This is another kind of weak primality.
3 x 11: Anomaly
5 x 7: twin primes
3 x 13: Anomaly
We revise the theory as follows. All odd numbers are prime unless the final digit is a 5. Hereafter in this research, numbers of this type will be disregarded.
7 x 7: Prime squared
3 x 19: Anomaly
7 x 9: Product of twin primes (9 being a weak prime)
3 x 23: Anomaly
7 x 11: Anomaly
9 x 9: Prime squared
3 x 29: Anomaly
7 x 13: Anomaly
3 x 31: Anomaly
9 x 11: Product of twin primes
Total anomalies: 9
Observation: All anomalies have a factor of 3 or 7. The theory allows exceptions for numbers divisible by 5. Since 5 is the mean value between 3 and 7, we may expect that the effects of divisibility by 3 or 7 will be offsetting, and therefore may be disregarded. In other words, divisibility by 3 or 7 is another case of weak primality and therefore not true anomalies. They are only apparent anomalies.
Summarizing so far . . .
Our theory asserts that any odd number is prime, with some exceptions for numbers with a final digit of 5. The theory also defines two kinds of prime numbers, strong and weak. Strong primes have no factors other than themselves and unity. Weak primes perfect squares of prime numbers, or else are products of twin primes. And, a number divisible by 3 or 7 may also be considered a weak prime.
The data we have gathered so far show the following for the odd numbers below 100:
There are only six exceptions under the digit-5 rule: 45, 55, 65, 75, 85 and 95, since 5 is a strong prime, while 15, 25 and 35 are weak primes. Counting them, we have found 18 weak primes, 25 strong primes, and no exceptions unaccounted for by the theory.
We might note that, of all numbers ending in 5, we have exhausted all potential weak primes, so we know that all the rest will be allowable exceptions and may be disregarded as far as whether they may confirm or disconfirm the theory.
For the numbers between 100 and 200, we present only a summary of our findings. Between 100 and 200 we find 21 strong primes, 18 weak primes and only one anomaly: 187 has 11 and 17 for factors.
We may reasonably assume that there is a logical explanation for this one apparent inconsistency between theory and observation. Science does not have all the answers and logic is not the source of all knowledge. The universe is full of mysteries. All humans must have faith in something because we all believe some things that we cannot prove.
Therefore, the exception proves the rule, and we are thus justified in believing that all odd numbers are prime.
Skeptics will naturally continue to look for evidence to the contrary. They will not accept the evidence presented here because they have made their minds up that there can be no evidence for this theory. No matter how thoroughly we might investigate this phenomenon, they will find fault. Our research could never satisfy them, because the primality of all odd numbers does not fit their world view. Their minds are made up. They will not be confused by facts.
We may expect that in their desperate efforts to discredit this theory, they will claim to have found odd numbers with all sorts of factors. This is typical of people who think they already have all the answers. They find what they expect to find. Those who want a reason not to believe can always find one.
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