4. What do you notice about the values of the separate square-, cube- and fourth-roots in all the questions above?

5. Look at the Table of relationships between Phi, phi and V5 and see if you can spot the two expressions in each of questions. So when we take the square-roots in question (1) and the cube-roots in question (2), and the fourth-roots in question (3), what are the results for each expression?

6. Finally, does it matter if we use different columns of figures for the two expressions in the trick?

Now you know the secret behind this trick!

With thanks to R. S. (Chuck) Tiberio of Wellesley, MA, USA for pointing out to me the basic relationships that this trick depends upon. He was one of the solvers of the original problem which you can find in:

Problem 402in The College Mathematics Journal, vol. 21, No. 4, September 1990, page 339. For a similar unlikely-looking collection of identities see:

^^^ Incredible Identities by D Shanks in Fibonacci Quarterly vol 12 (1974) pages 271 amd 280.

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

The Lucas Numbers in Pascal's Triangle

We found the Fibonacci numbers appearing as sums of "diagonals" in Pascal's Triangle on the Mathematical Patterns in the Fibonacci Numbers page. We can also find the Lucas numbers there too.

Here is the alternative form of Pascal's triangle that we referred to above, with the diagonals re-aligned as columns and the sums of the new columns are the Fibonacci numbers:

01 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 | |

0 |
1. | ||||||||

1 |
.1 |
1 | |||||||

2 |
1 |
2 |
1 | ||||||

3 |
1 |
3 |
3 |
1 | |||||

4 |
1 |
4 |
6 |
4 |
1 | ||||

5 |
1 |
5 |
10 |
10 |
5 | ||||

6 |
1 |
6 |
15 |
20 | |||||

7 |
1 |
7 |
21 | ||||||

8 |
1 |
8 |

To derive the Lucas numbers we still add the columns, but to each number in the column we first multiply by its column number and divide by its row number! Here's an example:-

Let's take the third column which, when after the appropriate multiplications and divisions should sum to L(3) which is 4. The lowest number in column 3 is 1 and it is on row 3, so we need:

which, in this case, doesn't alter the number by much!

The other number in column 3 is 2 on row 2, so this time we have:

Note that for all the numbers in the same column, we will always multiply by the same number - the column number is the same for all of them - but the divisors will alter each time.

Adding the numbers we have derived for this column we have 1+3=4 which is the third Lucas number L(3). Here is what happens in column 4, starting from the bottom again:-

Here's our revised Pascal's triangle from above showing some of the fractions that we use to derive the Lucas numbers - it shows the pattern in the multipliers and divisors more easily:

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 | |

0 |
1 | |||||||||

1 |
1x1/1=1 |
1x2/1=2 | ||||||||

2 |
1x2/2=1 |
2x3/2=3 |
1x4/2=2 | |||||||

3 |
1x3/3=1 |
3x4/3=4 |
3x5/3=5 |
1x6/3=2 | ||||||

4 |
1x4/4=1 |
4x5/4=5 |
6x6/4=9 |
4x7/4= 7 |
1x8/4= 2 | |||||

5 |
1x5/5=1 |
5x6/5=6 |
10x7/5=14 |
10x8/5=16 |
... | |||||

6 |
1x6/6=1 |
6x7/6= 7 |
15x8/6=20 |
... | ||||||

7 |
1x7/7= 1 |
7x8/7= 8 |
... | |||||||

8 |
1x8/8= 1 |
... | ||||||||

1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
... |

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

rrrrrrr rrrr^JJ

References

^ Lucas and Primality Testing Hugh C Williams, Wiley, 1998, ISBN: 0471 14852 0

is a new book (due April 1998) on how to test if a number is prime without factoring it using a technique developed by Edouard Lucas, with modern extensions to his work.

Primality testing has become a focus of modern number-theory and algorithmics research. Our present inability to find prime factors of a number in a fast and efficient way is relied upon in encryption systems - systems which encode information to send over phone lines. Such encryption systems are now built into computer chips in

• cash-card machines which communicate with your bank's central computing service to check your PIN and to verify the transaction;

• electronic cash transfer over the WWW where your browser encodes the message

• credit card transactions when your card is swiped through a machine at the till

Each of these systems must send the information in a secure way, free from tampering by fraudsters.

Fibonacci - the man and His

Times

Fibonacci Home Page ^ Fibonacci Forgeries!

^ The first 100 Lucas Numbers

^ The Golden Section In Art, Architecture and Music

Fibonacci, Phi and Lucas numbers Formulae

Links and References

Dr Ron Knott [email protected] Created: 18 October 1997 Updated: 7 May 2001

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