Pseudotheorem

In logic (especially in its applications to mathematics and philosophy), a pseudotheorem is a proposed general statement that is "coincidentally" true, rather than being the consequence of any axioms, postulates, or theorems, or any sequence of logical deductions thereof. For example, the statement, "All odd numbers between 0 and 14 are either prime or square" is a pseudotheorem because there is no "theoretical" reason it must be true, yet it is true as there are no counterexamples to the statement (1 and 9 are squares and 3, 5, 7, 11, and 13, are all prime). Many pseudotheorems are illustrative examples of the strong law of small numbers.

Examples

• Let x be a positive integer between 818 and 830, inclusive. Then, "x is prime if and only if x minus 810 is also prime" is a pseudotheorem. Each of the integers 821, 823, 827, and 829 are prime; in mathematical terms, we can say that they form a prime quadruplet. Subtracting 810 from all four numbers gives (11, 13, 17, 19), all of which are prime. One can then easily verify that there are no other primes in the interval between 818 and 830, nor are there any other primes in the interval between 8 and 20. As a result, the statement has no counterexamples and is thus a true statement.
• Let n be a positive integer such that n < 9. Then, $\lceil exp(n/2)\rceil$ = F_n+2, where exp denotes the exponential function eⁿ and F_n denotes the nth Fibonacci number. The two functions coincidentally and amusingly agree for the first eight terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. After that, the two functions diverge, with the function on the left side taking 91 and 149 for the ninth and tenth terms, respectively, while the function on the right side takes the values 89 and 144 for the ninth and tenth terms, respectively. As there are no counterexamples to the statement in the range between 0 and 8, inclusive, the statement holds for that range and is thus a pseudotheorem for said range.
• Let x be a positive integer such that x < 15. Then, x is either prime or square.
• Let x be a positive integer such that 115 < x < 125. Then, x is composite. See prime gap for more information.
• Let p be a prime number such that 1 < p < 10. Then, 2^p-1 is prime: 3, 7, 31, and 127 are all prime.
• Let n be a non-negative integer such that n < 6. Then, 30n+7 is prime: 7, 37, 67, 97, 127, and 157 are all prime. Unfortunately, the next three terms, 187, 217, and 247, are all composite (187 = 11 x 17, 217 = 7 x 31, and 247 = 13 x 19).

References

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