The Math Major Vol. 3, No. 12

The Math Major

CSU Fresno Mathematics Department

Vol 3. No. 12 (April 29, 1999)

Colloquium

Date: Friday May 7, 1:10pm
Location: Science 143
Speaker: Doug Burke, Department of Mathematics and Statistics, University of Nevada, Las Vegas
Title: "Truth and Provability"
Description: A mathematical statement is independent (of some collection of axioms) if the statement is neither provable nor refutable (from the axioms). In 1930 Godel shocked the mathematical and logical worlds by proving, under very mild conditions, that independent statements exist. This proof completely disrupted the traditional view of truth and falsehood in an axiomatic system- previously, the assumption was that mathematical statements were either provably true or provably false within that system. With the existence of independent statements first demonstrated by very contrived statements, an obvious question is whether or not there are naturally-occurring independent statements. After Paul Cohen's discovery of forcing in 1963, many natural mathematical problems have been found to be independent. We will present some examples of independence in cardinal arithmetic, and give some recent, surprising theorems in cardinal arithmetic, due to Shelah.

Eighteenth CSU, Fresno Math Field Day: A Success

On Saturday April 17, the CSUF math department hosted the Math Field Day for high schools. Students competed in competitions that required mathematical problem solving skills. The math department faculty would sincerely like to thank the CSU Fresno student volunteers and the staff of the department office who helped make the event possible.

Outstanding Math Majors

Amy Peterson, senior math major, was selected to be this years outstanding math major by the department faculty. Amy was also nominated by the department for the Rodman Presidential Fellowship for Entering Graduate Students. Math major Mindy Mallard was nominated by the department to be one of the recipients of the 1999-2000 F. Harold Downing scholarship.

Graduating Math Majors

There are 21 math majors and 2 masters degree (in mathematics) students graduating in the 1998-99 academic year (Fall and Spring). Many of our graduates choose teaching as a career. While the financial rewards in teaching are modest, the personal gratification teachers experience can be potent. The faculty would like to wish our former students well.

Graduation Dinner

Every Spring, the Department of Mathematics hosts a dinner for all graduating seniors and graduate students. This year the dinner will be held on Monday May 10. If you are a graduating math student you should have received an invitation. If you have not received an invitation, please contact the math department.

A Mathematical Quote

"I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling."

--Agatha Christie

Agatha Christie (1890-1978)

Problem Corner

Problem 3.11. Let k be a positive integer. Find the largest power of 3 which divides 10^k - 1.

Solution to Problem 3.11: If k = 3^m n, where n is not divisible by 3, we prove that 10^k-1 is divisible by 3^m+2, but not by 3^m+3.

The proof is by induction on m, and uses the fact that an integer is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. For m=0, k is not divisible by 3. We have 10^k-1 = 9x111...1, (k 1's), and the sum of the digits 111...1 is k, so 10^k-1 is divisible by 3², but not by 3³. We now assume the claim holds for m. Let k = 3^m+1n where n is not divisible by 3. Then 10^k-1 = (10^{3^m n})³ - 1 = (10^{3^m n} - 1)(10^{2x3^m n} + 10^{3^m n} + 1).

By the inductive hypothesis, the largest power of 3 dividing the first factor is 3^m+2. The sum of the digits of the second factor is 3, which is divisible by 3, but not by 3². Therefore, the highest power of 3 dividing 10^k-1 is 3^m+3, and the proof is complete.

There were no correct solutions turned in for this problem.

Problem Corner Winners for Spring 1999

John Jamison and Jon Klassen will share the $125 combined first and second prize for the Problem Corner contest this Spring semester. John and Jon turned in the only correct solutions to problem 3.11. No other solutions to the other problems (3.8, 3.9 and 3.11) were received.

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