Common intuition is a clandestine notion hidden behind the wall of mathematical formalism. Elaborate systems that dispose of specification in favor of complete abstraction are merely derived from basic understanding. Axiomatic truths build elegant mathematical structures layered by bricks of minor principles. In example, by using only three postulates the entire theory of Calculus can be constructed.

Oftentimes, mathematical axioms are simply self-evident truths that are drawn from a set of numbers. Number theorists, in particular, have found that numbers themselves exhibit their own special character. Indeed, in the field of Number Theory, numbers can be abundant, semiperfect, and weird. In fact, these strangely named numbers define a major unsolved problem.

An abundant number is defined as a number in which the sum of its proper divisors is greater than the number itself. Take the number 24 for example. Since {1,2,3,4,6,8,12} is the set of proper divisors ( all of the divisors except 24) of 24 and 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, 24 is an abundant number. If a natural number is the sum of all or some of its proper divisors, then it is a semiperfect number. Consequently, by inspection, a perfect number is a semiperfect number that is the sum of all of its proper divisors.

Considering these definitions, a weird number can be defined. Weird numbers are natural numbers that are abundant but not semiperfect. For example, 70 is the lowest weird number, because its set of proper divisors {1,2,5,7,10,14,35} sum to 74, but no subset of its set of proper divisors sum to the number 70, and 70 is the smallest number to meet such conditions. The first few weird numbers are 70, 836, 4030, and 5830 and so on.

In this small sampling of weird numbers, one can see that they are all even. Thus, the following question could be asked: Do any odd weird numbers exist? This is an easy question to ask, but far from easy to answer. However, other basic properties can be described with more ease. For instance, mathematicians have proven that there are infinitely many weird numbers. Also, if an odd weird number does exist, then it must be greater than .

Therefore, it is unlikely that pure computation alone will answer the question. Perhaps a solution is bound behind the wall of mathematical formalism. However, the answer could emerge from behind the bricks into the light of common understanding.

## About Avery Carr

I am a grad student studying mathematics at Emporia State University. I obtained a B.S. in Mathematics from the University of Memphis in 2010 where I served as President of the math club my senior year. My research interests include Graph Theory, Number Theory, Recreational Mathematics, Mathematical Physics, and Operator Theory.

You said that “by using only three postulates the entire theory of Calculus can be constructed”. What are these postulates?

Pedro,

Thank you for your comment. I first heard of this in a Real Analysis lecture the other day. Our class is using the text An Introduction To Analysis by William R. Wade. My professor said “that some texts show that Calculus can be derived entirely by five postulates, but our text does it with three.” The postulates are the following:

Postulate 1 : (Field Axioms) Given addition and multiplication on R x R (can be shown for complex numbers as well) then by showing:

Closure, Associative, Commutative, Distributive, Existence of the Additive Identity, Existence of the Multiplicative Identity, Existence of Additive Inverse,

we have shown R ( could do it of complex numbers as well) is a field.

Postulate 2: (Order Axioms) Trichotomy Property, Transitive Property, Additive Property, and Multiplicative Property for the partial order relation < on R x R.

Postulate 3: (Completeness Axiom). If E is a nonempty subset of R that is bounded above, then E has a finite supremum.

So, in not taking what my professor said on faith, I justified in my mind how these postulates really do form a basis for Calculus. Thanks again.