Saturday, October 23, 2010

More Fun With Num3ers: Calculate your age in days

More Fun With Num3ers: Calculate your age in days: "You can calculate your age in days at ... http://www.derbyshireguide.co.uk/young.htm (1) Is your number a prime number or a composite number..."

Calculate your age in days

You can calculate your age in days at ...


(1) Is your number a prime number or a composite number?

Here's a list of the First 10,000 Prime Numbers


A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself.

A composite number is a positive integer which has a positive divisor other than one or itself.

Example:
9 can be divided evenly by 1, 3 and 9. So 9 is a composite number.
7 cannot be divided up evenly. we could have divided 7 into seven 1s, or one 7

(2) Is your number a square? cubic? or something else?


Friday, October 22, 2010

More Fun With Num3ers: Waring's problem for cubes

More Fun With Num3ers: Waring's problem for cubes: "Waring's problem for cubes : Every positive integer can be written as the sum of nine (or fewer) positive cubes. 6 = 1^3 + 1^3 + 1^3 + 1^3 ..."

Thursday, October 21, 2010

Waring's problem for cubes

Waring's problem for cubes : Every positive integer can be written as the sum of nine (or fewer) positive cubes.


6 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
7 = 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
8 = 2^3
9 = 2^3 + 1^3
10 = 2^3 + 1^3 + 1^3
11 = 2^3 + 1^3 + 1^3 + 1^3
12 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3
13 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
14 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
15 = 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3
23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.
24 = 2^3 + 2^3 + 2^3
29 = 3^3 + 1^3 + 1^3
31 = 3^3 + 1^3 + 1^3 + 1^3 + 1^3
51 = 3^3 + 2^3 + 2^3 + 2^3

Based on this, is it reasonable to suppose that every integer is the sum of nine positive cubes (or fewer) ?

23 is written as the sum of nine positive cubes. Can you find other numbers which can be written as the sum of nine positive cubes?




More Fun With Num3ers: Integers that are both perfect squares and cubes

More Fun With Num3ers: Integers that are both perfect squares and cubes: "64 is the first number after 1 to be both a square and a cube: 64 = 8^2 = 4^3. Let's find few others. The On-Line Encyclopedia of Integer Se..."

Integers that are both perfect squares and cubes

64 is the first number after 1 to be both a square and a cube: 64 = 8^2 = 4^3.

Let's find few others.

The On-Line Encyclopedia of Integer Sequences gives


0, 1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 16777216, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 244140625, 308915776

0 = 0^3 = 0^2
1 = 1^3 = 1^2
64 = 4^3 = 8^2
729 = 9^3 = 27^2
4096 = 16^3 = 64^2
15625 = 25^3 = 125^2
46656 = 36^3 = 216^2
117649 = 49^3 = 343^2
etc.

A pattern emerges..... Do you see it?




Tuesday, October 19, 2010

More Fun With Num3ers: More Fun Facts With Num3ers

More Fun With Num3ers: More Fun Facts With Num3ers: "1 + 3 + 1 = 1^2 + 2^21 + 3 + 5 + 3 + 1 = 2^2 + 3^21 + 3 + 5 + 7 + 5 + 3 + 1 = 3^2 + 4^2, etc... 153 is the sum of the first five factorials..."

More Fun Facts With Num3ers

1 + 3 + 1 = 1^2 + 2^2
1 + 3 + 5 + 3 + 1 = 2^2 + 3^2
1 + 3 + 5 + 7 + 5 + 3 + 1 = 3^2 + 4^2,
etc...

153 is the sum of the first five factorials: 1! + 2! + 3! + 4! + 5! = 153

The 26 letters of the alphabet can make 40,329 * 10^22 different combinations

1961 was the most recent year that could be written upside-down and rightside-up and appear the same. The next year with the same properties is 6009

If you change a $5 bill into all the possible ways (cents,nickels,dimes,quarters,halves and dollars) it would require: 2,305,843,009,213,693,951 different changes

Recreation with the 8x table:
1*8=8 (8);
2*8=16 (1+6=7);
3*8=24 (2+4=6);
4*8=32 (3+2=5);
5*8=40 (4+0=4);
6*8=48 (4+8=12, 1+2=3);
7*8=56 (5+6=11, 1+1=2) ...

11264 = 11 * 2^{6+4} .............. [All digits of the number 11264 are used]

2737 = (2 × 7)^3 – 7 ................. [idem]


Royal V. Heath discovered 1118 + 1881 + 8181 + 8818 = 1181 + 1818 + 8118 + 8881

Reverse that and you get:
1888 + 8118 + 8181 + 1811 = 8188 + 1818 + 1881 + 8111

Both equations remain true if you square all their terms.