Number Theory and Algebra Working Group

Tuesday, July 10

"Everything is the thing of the sum of up and over"

A. Cuoco

Recall that if f is a function from W to W (from the counting numbers to the counting numbers), then f determines a sequences of “differences” Delta (n) where Delta (n) = f (n+1) – f (n).

For example, here’s a partial table of a function.

Values of a certain function f(n)

n

f(n)

Delta(n)  = f(n+1)- f(n)

0

3

 

1

8

5

2

13

5

3

18

5

4

23

5

5

28

5

6

33

5

We determined that the function is f (n) = 3n+1 (assuming that Delta (n) remains constantly 5).

Here’s another example. Observe that Delta Delta is the “second difference”, the sequence which record s the differences of the differences, if you will.

n

g(n)

Delta(n)

Delta Delta(n)

0

3

1

3

1

4

4

3

2

8

7

3

3

15

10

3

4

25

13

3

5

38

 

 

Last night, Gary proved the following:

Theorem If f is a function from W to W such that the second differences are constant, then f is a quadratic function, i.e. f (n) = ax^2 + bx   + c, for some real numbers a, b, c.

Gary provided a proof during the meeting today.

Let’s look at the function g(n) again.  Observe that Delta (n) is a linear function (since the first differences for Delta (n) are constant. In fact, Delta (n) = 3n +1.

n

g(n)

Delta(n)

Delta Delta(n)

0

3

1

3

1

4

4

3

2

8

7

3

3

15

10

3

4

25

13

3

5

38

 

 

So how do we determine g (n) from Delta (n)?  Observe that g (1) = g (0) + Delta (0),

g (2) =  g(1) + Delta(1)  = g(0) + Delta(0) + Delta(1). Hey, see a pattern developing here?

In general, g (n) = g (0) + Delta (0) + Delta (1) +.,,+ Delta(n-1).

In Gary’s proof, the main idea was to show that if Delta (n) is linear, then the function g (n) is quadratic.

Amazingly we were able to show that from the first line of a table for a function h (x) with a constant second difference, if the first line is

n

h(n)

Delta(n)

Delta Delta(n)

0

r

s

t

(where r,s,t are numbers)

then h(n) =  r + sn + t(n(n-1)/2) =  (t/2)n^2 + (s- t/2)n + r

We then discussed summation notation, how Pascal’s Triangle is lurking in the background, We had further discussion concerning binomial coefficients, the binomial theorem- we will discuss these ideas further.

“I thought I did but now I don’t” John

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.