You are here:

Advanced Math/proof of primes

Advertisement

Expert: - 5/5/2009


Question
QUESTION: Hi Scott,
Could you assist me in proving the following?
Prove or Disprove: P is greater or equal to 5 is a prime number if and only if P is one more or one less than a multiple of six.
Thanks,
Evelyn

ANSWER: The possible list of all numbers that are greater than 5:
6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, and 6n + 5 for any integer n>=1.

For 6n, the number is divisible by 6.

For 6n + 2, the number is the same as 2(3n+1),
so the number is divisible by 2.

For 6n + 3, the number is 3(n+2), so it is divisble by 3.

For 6n + 4, the number is divisible by 2, for it's 2(3n + 2).

The only ones that have not been discussed of 6n + 1 and 6n + 5.
Note that 6n + 5 = 6n + 6 - 1 = 6(n+1) - 1.  This shows that the only prime numbers are 1 more or 1 less than a multiple of 6.


---------- FOLLOW-UP ----------

QUESTION: Scott,
Thank you so much for your answer. Although I am very passionate about math, I am am also very new at discrete mathematics. I really liked how you broke everything down.  Let me ask you... when writing a proof, would I do proceed directly by using the first examples then indirectly using 6n+1 and 6n+5?  In other words, assume q and then not q?  If so, we assume P is one more or one less than a multiple of six and then assume P IS NOT one mor or one less than a multiple of six?  Is that how it works?  Please advice.  Evelyn

Answer
Start by assuming all numbers to be prime (we both know they're not, but just do it anyway).

The numbers start at 5, which is 6-1.
The numbers would then be 6n - 1, 6n, 6n + 1, 6n + 2, 6n + 3,
6n + 4, and 6n + 5 for n chosen out of the positive integers.

As was shown in the last letter,

For 6n, the number is divisible by 6.

For 6n + 2, the number is the same as 2(3n+1),
so the number is divisible by 2.

For 6n + 3, the number is 3(n+2), so it is divisble by 3.

For 6n + 4, the number is divisible by 2, for it's 2(3n + 2).

Therefore all numbers of the form 6n, 6n + 2, 6n + 3, and 6n + 4
are not prime.  The only possibilities this leaves are
6n - 1 and 6n + 1.

Advanced Math

All Answers

Answers by Expert:

Ask Experts

Volunteer

Scott A Wilson

Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

Experience

Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people nearby, friends with math questions, and several people every day on the PC, and you're probably make one more.

©2012 About.com, a part of The New York Times Company. All rights reserved.