Almost four years ago my friend, Dave Renfro, sent me a packet of papers from his searching in the archives of old journals. I was on vacation at the time and on my return somehow put t hem away in the stacks beneath my library area. Four years later my wife was organizing my life as she occasionally does, and pulled them out for me to see if they were to keep, or throw out.
The first article I pulled out of the thick packet was written about the time I was just being introduced to Algebra, in 1957 in the Mathematics Magazine. It was written by a Doyne Holder from Kinkaid School, in Houston, Tx. Amazingly, I spent a pretty full lifetime of studying and teaching mathematics and never had this insight, so it may also be true of other teachers. For them, walking in my footsteps, I share this beautiful little gem:
If a Quadratic of the form ax2 + bx + c, if all three coefficients a, b, and c are odd, then the quadratic is not factorable over the integers.
The proof is as simple as the ideas of quadratic multiplication. Assume the factors are mx+r and nx+s, then it must be true that
a=m*n and
c= r*s
And b = m*s + n*r, and since all of m,n,r, and s are odd, m*s is odd, and n*r is odd, but the sum of two odds can not be odd, and so b can not be odd.
He went on to show that for a larger degree polynomial, a similar proof exists that if the coefficient of the highest power, as well as the constant term and the linear coefficient are all odd, it is also impossible to factor the polynomial over the integers. This I will leave for the reader to enjoy the self discovery. If anyone gets terribly stuck I can supply a not too subtle clue, and if needed, the complete very short proof.
And now to dive into the rest of Dave's packet.... no treasure like an old math paper rediscovered.
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