## Saturday, January 29, 2011

### Fermat ``near-misses''

Tables of Fermat "near-misses''

### Clock puzzles

Here are two clock puzzles invented by Lewis Carroll:

(1) A clock has hour and minute hands of the same length and no numerals on its face. At what time between 6 and 7 o'clock will the time on the clock appear to be the same as the time read on the reflection of the clock in a mirror?

(2) Which has a better chance of giving the right time: a clock that has stopped or one that loses a minute every day?

Here is another from Henry Dudeney's Amusements in Mathematics called "The Club Clock:"

One of the big clocks in the Cogitators' Club was found the other night to have stopped just when, as will be seen in the illustration, the second hand was exactly midway between the other two hands. One of the members proposed to some of his friends that they should tell him the exact time when (if the clock had not stopped) the second hand would next again have been midway between the minute hand and the hour hand. Can you find the correct time that it would happen?

Solutions
(1) 360/13 minutes after 6
(2) The stopped clock, because it will give the right time twice a day whereas the other is only correct about every two years approximately.

**************************

A well-known and simple puzzle is this: The hour and minute hands of a clock are superimposed at 12:00. When will they next be superimposed (I don't mean lined up as they are at 6:00)?

## Friday, January 28, 2011

### Pizza Cutting Problem: Dividing the plane

Suppose n straight cuts are made across a pizza.
What is the maximum of pieces, H_n, which can result?

### Pizza Cutting Problem (Challenging Problem)

Theorem :

There is an algorithm that, given a cutting of the pizza with n slices, performs a precomputation in time O(n). Then, during the game, the algorithm decides each of Alice’s turns in time O(1) in such a way that Alice makes at most two jumps and her gain is at least g(n)|P|.

Claim :

There is an algorithm that, given a cutting of the pizza with n slices, computes an optimal strategy for each of the two players in time O(n^2). The algorithm stores an optimal turn of the player on turn for all the n^2−n+2 possible positions of the game.

Open problem

Is there an algorithm that uses o(n^2) time for some precomputations and then computes each optimal turn in constant time?

See details

## Wednesday, January 26, 2011

Steven G. Krantz

### Seven Prime Questions

http://www.pbs.org/wgbh/nova/physics/prime-questions.html

In this audio interactive, five of the mathematicians who meet each week for the New York Number Theory Seminar answer (a prime number) of prime number questions, including: Why are prime numbers fascinating? and When will there be a final solution to the twin prime conjecture?

Printable version

Twin Prime Conjecture

## Tuesday, January 25, 2011

### Fun with the language

Victor Borge - Inflationary Language

Victor Borge does the Inflationary language. He swiches every number in various words, for one number higher and then reads a text in this manner.

The Victor Borge Website Inflater

List all words that contain zero
http://www.morewords.com/contains/zero/

List all words that contain one

List all words that contain two

List all words that contain three
http://www.morewords.com/contains/three/

List all words that contain four

List all words that contain five
http://www.morewords.com/contains/five/

List all words that contain six

List all words that contain seven

List all words that contain eight

List all words that contain nine

List all words that contain ten

### Geometry for all ages

Older posts

Eudoxus

Heart Curve

Bonne Projection

Bring Geometry to Life with Google SketchUp

## Monday, January 24, 2011

### Palindromic numbers

Numbers n such that n^2+n+2 is a palindrome

## Sunday, January 23, 2011

### Prime number abc... such that a^a + b^b + c^c + ... is prime

The First 1,000 Primes

2 ..... 3..... 5 ..... 7 ..... 11 ..... 13 ..... 17 ..... 19 ..... 23 ..... 29 ..... 31 ..... 37 ..... 41 .....
43 ..... 47 ..... 53 ..... 59 ..... 61 .... 67 ..... 71 ..... 73 ..... 79 ..... 83 ..... 89 ..... 97 .....
101 ... 103 ... 107 ... 109 ... 113 ... 127 ... 131 ... 137 ... 139 ... 149 ... 151 ... 157 ... 163 ... 167 ... 173 ... 179 ... 181 ... 191 ... 193 ... 197 ... 199 ... 211 ... 223 ... 227 ... 229 ... 233 ... 239 ... 241 ... 251 ... 257 ... 263 ... 269 ... 271 ... 277 ... 281 ... 283 ... 293 ... 307 ... 311 ... 313 ... 317 ... 331 ... 337 ... 347 ... 349 ... 353 ... 359 ... 367 ... 373 ... 379 ... 383 ... 389 ... 397 ... 401 ... 409 ... 419 ... 421 ... 431 ... 433 ... 439 ... 443 ... 449 ... 457 ... 461 ... 463 ... 467 ... 479 ... 487 ... 491 ... 499 ... 503 ... 509 ... 521 ... 523 ... 541 ... 547 ... 557 ... 563 ... 569 ... 571 ... 577 ... 587 ... 593 ... 599 ... 601 ... 607 ... 613 ... 617 ... 619 ... 631 ... 641 ... 643 ... 647 ... 653 ... 659 ... 661 ... 673 ... 677 ... 683 ... 691 ... 701 ... 709 ... 719 ... 727 ... 733 ... 739 ... 743 ... 751 ... 757 ... 761 ... 769 ... 773 ... 787 ... 797 ... 809 ... 811 ... 821 ... 823 ... 827 ... 829 ... 839 ... 853 ... 857 ... 859 ... 863 ... 877 ... 881 ... 883 ... 887 ... 907 911 ... 919 ... 929 ... 937 ... 941 ... 947 ... 953 ... 967 ... 971 ... 977 ... 983 ... 991 ... 997 ... 1009 ... 1013 ... 1019 ... 1021 ... 1031 ... 1033 ... 1039 ... 1049 ... 1051 ... 1061 ... 1063 ... 1069 ... 1087 ... 1091 ... 1093 ... 1097 ... 1103 ... 1109 ... 1117 ... 1123 ... 1129 ... 1151 ... 1153 ... 1163 ... 1171 ... 1181 ... 1187 ... 1193 ... 1201 ... 1213 ... 1217 ... 1223 ... 1229 ... 1231 ... 1237 ... 1249 ... 1259 ... 1277 ... 1279 ... 1283 ... 1289 ... 1291 ... 1297 ... 1301 ... 1303 ... 1307 ... 1319 ... 1321 ... 1327 ... 1361 ... 1367 ... 1373 ... 1381 ... 1399 ... 1409 ... 1423 ... 1427 ... 1429 ... 1433 ... 1439 ... 1447 ... 1451 ... 1453 ... 1459 ... 1471 ... 1481 ... 1483 ... 1487 ... 1489 ... 1493 ... 1499 ... 1511 ... 1523 ... 1531 ... 1543 ... 1549 ... 1553 ... 1559 ... 1567 ... 1571 ... 1579 ... 1583 ... 1597 ... 1601 ... 1607 ... 1609 ... 1613 ... 1619 ... 1621 ... 1627 ... 1637 ... 1657 ... 1663 ... 1667 ... 1669 ... 1693 ... 1697 ... 1699 ... 1709 ... 1721 ... 1723 ... 1733 ... 1741 ... 1747 ... 1753 ... 1759 ... 1777 ... 1783 ... 1787 ... 1789 ... 1801 ... 1811 ... 1823 ... 1831 ... 1847 ... 1861 ... 1867 ... 1871 ... 1873 ... 1877 ... 1879 ... 1889 ... 1901 ... 1907 ... 1913 ... 1931 ... 1933 ... 1949 ... 1951 ... 1973 ... 1979 ... 1987 ... 1993 ... 1997 ... 1999 ... 2003 ... 2011 ... 2017 ... 2027 ... 2029 ... 2039 ... 2053 ... 2063 ... 2069 ... 2081 ... 2083 ... 2087 ... 2089 ... 2099 ... 2111 ... 2113 ... 2129 ... 2131 ... 2137 ... 2141 ... 2143 ... 2153 ... 2161 ... 2179 ... 2203 ... 2207 ... 2213 ... 2221 ... 2237 ... 2239 ... 2243 ... 2251 ... 2267 ... 2269 ... 2273 ... 2281 ... 2287 ... 2293 ... 2297 ... 2309 ... 2311 ... 2333 ... 2339 ... 2341 ... 2347 ... 2351 ... 2357 ... 2371 ... 2377 ... 2381 ... 2383 ... 2389 ... 2393 ... 2399 ... 2411 ... 2417 ... 2423 ... 2437 ... 2441 ... 2447 ... 2459 ... 2467 ... 2473 ... 2477 ... 2503 ... 2521 ... 2531 ... 2539 ... 2543 ... 2549 ... 2551 ... 2557 ... 2579 ... 2591 ... 2593 ... 2609 ... 2617 ... 2621 ... 2633 ... 2647 ... 2657 ... 2659 ... 2663 ... 2671 ... 2677 ... 2683 ... 2687 ... 2689 ... 2693 ... 2699 ... 2707 ... 2711 ... 2713 ... 2719 ... 2729 ... 2731 ... 2741 ... 2749 ... 2753 ... 2767 ... 2777 ... 2789 ... 2791 ... 2797 ... 2801 ... 2803 ... 2819 ... 2833 ... 2837 ... 2843 ... 2851 ... 2857 ... 2861 ... 2879 ... 2887 ... 2897 ... 2903 ... 2909 ... 2917 ... 2927 ... 2939 ... 2953 ... 2957 ... 2963 ... 2969 ... 2971 ... 2999 ... 3001 ... 3011 ... 3019 ... 3023 ... 3037 ... 3041 ... 3049 ... 3061 ... 3067 ... 3079 ... 3083 ... 3089 ... 3109 ... 3119 ... 3121 ... 3137 ... 3163 ... 3167 ... 3169 ... 3181 ... 3187 ... 3191 ... 3203 ... 3209 ... 3217 ... 3221 ... 3229 ... 3251 ... 3253 ... 3257 ... 3259 ... 3271 ... 3299 ... 3301 ... 3307 ... 3313 ... 3319 ... 3323 ... 3329 ... 3331 ... 3343 ... 3347 ... 3359 ... 3361 ... 3371 ... 3373 ... 3389 ... 3391 ... 3407 ... 3413 ... 3433 ... 3449 ... 3457 ... 3461 ... 3463 ... 3467 ... 3469 ... 3491 ... 3499 ... 3511 ... 3517 ... 3527 ... 3529 ... 3533 ... 3539 ... 3541 ... 3547 ... 3557 ... 3559 ... 3571 ... 3581 ... 3583 ... 3593 ... 3607 ... 3613 ... 3617 ... 3623 ... 3631 ... 3637 ... 3643 ... 3659 ... 3671 ... 3673 ... 3677 ... 3691 ... 3697 ... 3701 ... 3709 ... 3719 ... 3727 ... 3733 ... 3739 ... 3761 ... 3767 ... 3769 ... 3779 ... 3793 ... 3797 ... 3803 ... 3821 ... 3823 ... 3833 ... 3847 ... 3851 ... 3853 ... 3863 ... 3877 ... 3881 ... 3889 ... 3907 ... 3911 ... 3917 ... 3919 ... 3923 ... 3929 ... 3931 ... 3943 ... 3947 ... 3967 ... 3989 ... 4001 ... 4003 ... 4007 ... 4013 ... 4019 ... 4021 ... 4027 ... 4049 ... 4051 ... 4057 ... 4073 ... 4079 ... 4091 ... 4093 ... 4099 ... 4111 ... 4127 ... 4129 ... 4133 ... 4139 ... 4153 ... 4157 ... 4159 ... 4177 ... 4201 ... 4211 ... 4217 ... 4219 ... 4229 ... 4231 ... 4241 ... 4243 ... 4253 ... 4259 ... 4261 ... 4271 ... 4273 ... 4283 ... 4289 ... 4297 ... 4327 ... 4337 ... 4339 ... 4349 ... 4357 ... 4363 ... 4373 ... 4391 ... 4397 ... 4409 ... 4421 ... 4423 ... 4441 ... 4447 ... 4451 ... 4457 ... 4463 ... 4481 ... 4483 ... 4493 ... 4507 ... 4513 ... 4517 ... 4519 ... 4523 ... 4547 ... 4549 ... 4561 ... 4567 ... 4583 ... 4591 ... 4597 ... 4603 ... 4621 ... 4637 ... 4639 ... 4643 ... 4649 ... 4651 ... 4657 ... 4663 ... 4673 ... 4679 ... 4691 ... 4703 ... 4721 ... 4723 ... 4729 ... 4733 ... 4751 ... 4759 ... 4783 ... 4787 ... 4789 ... 4793 ... 4799 ... 4801 ... 4813 ... 4817 ... 4831 ... 4861 ... 4871 ... 4877 ... 4889 ... 4903 ... 4909 ... 4919 ... 4931 ... 4933 ... 4937 ... 4943 ... 4951 ... 4957 ... 4967 ... 4969 ... 4973 ... 4987 ... 4993 ... 4999 ... 5003 ... 5009 ... 5011 ... 5021 ... 5023 ... 5039 ... 5051 ... 5059 ... 5077 ... 5081 ... 5087 ... 5099 ... 5101 ... 5107 ... 5113 ... 5119 ... 5147 ... 5153 ... 5167 ... 5171 ... 5179 ... 5189 ... 5197 ... 5209 ... 5227 ... 5231 ... 5233 ... 5237 ... 5261 ... 5273 ... 5279 ... 5281 ... 5297 ... 5303 ... 5309 ... 5323 ... 5333 ... 5347 ... 5351 ... 5381 ... 5387 ... 5393 ... 5399 ... 5407 ... 5413 ... 5417 ... 5419 ... 5431 ... 5437 ... 5441 ... 5443 ... 5449 ... 5471 ... 5477 ... 5479 ... 5483 ... 5501 ... 5503 ... 5507 ... 5519 ... 5521 ... 5527 ... 5531 ... 5557 ... 5563 ... 5569 ... 5573 ... 5581 ... 5591 ... 5623 ... 5639 ... 5641 ... 5647 ... 5651 ... 5653 ... 5657 ... 5659 ... 5669 ... 5683 ... 5689 ... 5693 ... 5701 ... 5711 ... 5717 ... 5737 ... 5741 ... 5743 ... 5749 ... 5779 ... 5783 ... 5791 ... 5801 ... 5807 ... 5813 ... 5821 ... 5827 ... 5839 ... 5843 ... 5849 ... 5851 ... 5857 ... 5861 ... 5867 ... 5869 ... 5879 ... 5881 ... 5897 ... 5903 ... 5923 ... 5927 ... 5939 ... 5953 ... 5981 ... 5987 ... 6007 ... 6011 ... 6029 ... 6037 ... 6043 ... 6047 ... 6053 ... 6067 ... 6073 ... 6079 ... 6089 ... 6091 ... 6101 ... 6113 ... 6121 ... 6131 ... 6133 ... 6143 ... 6151 ... 6163 ... 6173 ... 6197 ... 6199 ... 6203 ... 6211 ... 6217 ... 6221 ... 6229 ... 6247 ... 6257 ... 6263 ... 6269 ... 6271 ... 6277 ... 6287 ... 6299 ... 6301 ... 6311 ... 6317 ... 6323 ... 6329 ... 6337 ... 6343 ... 6353 ... 6359 ... 6361 ... 6367 ... 6373 ... 6379 ... 6389 ... 6397 ... 6421 ... 6427 ... 6449 ... 6451 ... 6469 ... 6473 ... 6481 ... 6491 ... 6521 ... 6529 ... 6547 ... 6551 ... 6553 ... 6563 ... 6569 ... 6571 ... 6577 ... 6581 ... 6599 ... 6607 ... 6619 ... 6637 ... 6653 ... 6659 ... 6661 ... 6673 ... 6679 ... 6689 ... 6691 ... 6701 ... 6703 ... 6709 ... 6719 ... 6733 ... 6737 ... 6761 ... 6763 ... 6779 ... 6781 ... 6791 ... 6793 ... 6803 ... 6823 ... 6827 ... 6829 ... 6833 ... 6841 ... 6857 ... 6863 ... 6869 ... 6871 ... 6883 ... 6899 ... 6907 ... 6911 ... 6917 ... 6947 ... 6949 ... 6959 ... 6961 ... 6967 ... 6971 ... 6977 ... 6983 ... 6991 ... 6997 ... 7001 ... 7013 ... 7019 ... 7027 ... 7039 ... 7043 ... 7057 ... 7069 ... 7079 ... 7103 ... 7109 ... 7121 ... 7127 ... 7129 ... 7151 ... 7159 ... 7177 ... 7187 ... 7193 ... 7207 ... 7211 ... 7213 ... 7219 ... 7229 ... 7237 ... 7243 ... 7247 ... 7253 ... 7283 ... 7297 ... 7307 ... 7309 ... 7321 ... 7331 ... 7333 ... 7349 ... 7351 ... 7369 ... 7393 ... 7411 ... 7417 ... 7433 ... 7451 ... 7457 ... 7459 ... 7477 ... 7481 ... 7487 ... 7489 ... 7499 ... 7507 ... 7517 ... 7523 ... 7529 ... 7537 ... 7541 ... 7547 ... 7549 ... 7559 ... 7561 ... 7573 ... 7577 ... 7583 ... 7589 ... 7591 ... 7603 ... 7607 ... 7621 ... 7639 ... 7643 ... 7649 ... 7669 ... 7673 ... 7681 ... 7687 ... 7691 ... 7699 ... 7703 ... 7717 ... 7723 ... 7727 ... 7741 ... 7753 ... 7757 ... 7759 ... 7789 ... 7793 ... 7817 ... 7823 ... 7829 ... 7841 ... 7853 ... 7867 ... 7873 ... 7877 ... 7879 ... 7883 ... 7901 ... 7907 ... 7919

For example, 11 --> 1^1 + 1^1 = 2 is a prime number

13 --> 1^1 + 3^3 = 28 (= 2 * 2 * 7) is not a prime number
17 --> 1^1 + 7^7 = 823544 (= 2 * 2 * 2 * 113 * 911) not a prime number
19 --> 1^1 + 9^9 = 387420490 (= 2 * 5 * 73 * 530713)
23 --> 2^2 + 3^3 = 31 is a prime number
29 --> 2^2 + 9^9 = 387420493 (= 8353 * 46381)
31 --> 3^3 + 1^1 = 28 (just the reverse of 13)
37 --> 3^3 + 7^7 = 823570 (= 2 * 5 * 11 * 7487)
41 --> 4^4 + 1^1 = 257 is a prime number
43 ---> 4^4 + 3^3 = 283 is a prime number
47 ---> 4^4 + 7^7 = 823799 is a prime number

53 ---> 5^5 + 3^3 = 3152 = 2^4 * 197
59 ---> 5^5 + 9^9 = 387423614 = 2 * 67 * 409 * 7069
61 ---> 6^6 + 1^1 = 46657 = 13 * 37 * 97
67 ---> 6^6 + 7^7 = 870199 = 11 * 239 * 331
71 ---> 7^7 + 1^1 = 823544 = 2^3 * 113 * 911
73 ---> 7^7 + 3^3 = 823570 = 2 * 5 * 11 * 7487
79 ---> 7^7 + 9^9 = 388244032 = 2^6 * 11 * 551483
83 ---> 8^8 + 3^3 = 16777243 = 7 * 37 * 211 * 307
89 ---> 8^8 + 9^9 = 404197705 = 5 * 197 * 410353
97 ---> 9^9 + 7^7 = 388244032 = 2^6 * 11 * 551483

I will ignore prime numbers that contain the number 0, such 101, 103, 107, 109, etc.
since 0^0 is not defined.

### POLL: zero to the power zero

POLL: zero to the power zero is undefined. But if it could be defined,
what "should" it be? State your reasons.

Discussion: Zero to the zero power

### Pythagoras: Fun facts

Numbers to the Pythagorians were like the ideal forms sought after by Plato. Everything could be explained by their magic properties.

According to Pythagoras, numbers had a real and separate existence outside our minds.

10 was a perfect number, according to Pythagoras.

http://home.c2i.net/greaker/comenius/9899/pythagoras/pythagoras.html

10 is a triangular number: 1 + 2 + 3 + 4 = 10
10 is the only triangular number which is also the sum of 2
1^2 + 3^2 = 10 (consecutive square odd numbers)
10 cannot be the difference of 2 squares, because 10 is of the form 4n + 2
10 = 2 + 3 + 5 (sum of the first 3 primes)
10! = 6! * 7! = 3! * 5! * 7!
(Unique solution to the factorial equation n! = a! * b! * c! with consecutive prime factors)

(10!)^2 + 1 is a prime number

Pythagorean triples
http://en.wikipedia.org/wiki/Pythagorean_triple

A little fun with Pythagorean triples:

Exactly four right triangles with integer side lengths exist with a leg equal to 15. These are:

(15, 112, 113), (15, 20, 25), (15, 36, 39), and (8, 15, 17).

How many different right triangle with integer side lengths exist with a leg equal to 42? How do you know?

BONUS QUESTION:

42 is the product of 3 different prime numbers - 2, 3, and 7.

Resolve this problem given a leg which is the product of n different primes, given that, as in this problem, one of these primes is 2.

## Saturday, January 22, 2011

### Primes (abc)/(a*b*c) and (abcd)/(a*b*c*d)

(abc)/(a*b*c) is prime:

115 is the smallest : 115/5 = 23 is a prime number

(abcd)/(a*b*c*d) is prime:

1115 / 5 = 223 is a prime number

(abcde)/(a*b*c*d*e) is prime:

However 11115 / 5 = 2223 = 3 * 3 * 13 * 19

### Number Theory and Combinatorics articles

http://algo.inria.fr/bsolve/constant/constant.html

Function calculator

The Prime pages

Le Laboratoire de combinatoire et d'informatique mathématique (LaCIM)

### Palindromic numbers n(n + a)

To find n such that n(n + a) is a Palindromic number
a = 1, 2, 3, 4, 5, 6, 7, 8, 9

n(n + 1)

n(n + 2)

n(n + 3)

n(n + 4)

n(n + 5) ... 814 * (814 + 5) = 666666.

n(n + 6)

n(n + 7)

n(n + 8)

n(n + 9)

n(n + 20) ... 23 * (23 + 20) = 989

## Friday, January 21, 2011

### Cake numbers

Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3)+n+1

1, 2, 4, 8, 15, 26, 42, 64, 93, 130, ....

### 3-digit prime numbers such that

How many 3-digit prime numbers "abc" such that

a*b + c, a + b*c, and a + b + c are prime numbers, can you find?

Here's the list of 3-digit prime numbers:

101 ... 103 ... 107 ... 109 ... 113 ... 127 ... 131 ... 137 ... 139 ... 149 ... 151 ... 157 ... 163 ... 167 ... 173 ... 179 ... 181 ... 191 ... 193 ... 197 ... 199 ... 211 ... 223 ... 227 ... 229 ... 233 ... 239 ... 241 ... 251 ... 257 ... 263 ... 269 ... 271 ... 277 ... 281 ... 283 ... 293 ... 307 ... 311 ... 313 ... 317 ... 331 ... 337 ... 347 ... 349 ... 353 ... 359 ... 367 ... 373 ... 379 ... 383 ... 389 ... 397 ... 401 ... 409 ... 419 ... 421 ... 431 ... 433 ... 439 ... 443 ... 449 ... 457 ... 461 ... 463 ... 467 ... 479 ... 487 ... 491 ... 499 ... 503 ... 509 ... 521 ... 523 ... 541 ... 547 ... 557 ... 563 ... 569 ... 571 ... 577 ... 587 ... 593 ... 599 ... 601 ... 607 ... 613 ... 617 ... 619 ... 631 ... 641 ... 643 ... 647 ... 653 ... 659 ... 661 ... 673 ... 677 ... 683 ... 691 ... 701 ... 709 ... 719 ... 727 ... 733 ... 739 ... 743 ... 751 ... 757 ... 761 ... 769 ... 773 ... 787 ... 797 ... 809 ... 811 ... 821 ... 823 ... 827 ... 829 ... 839 ... 853 ... 857 ... 859 ... 863 ... 877 ... 881 ... 883 ... 887 ... 907 ... 911 ... 919 ... 929 ... 937 ... 941 ... 947 ... 953 ... 967 ... 971 ... 977 ... 983 ... 991 ... 997 ...

### Sports Betting: Billy Walters

Las Vegas sports betting legend Bill Walters has never had a losing year - a winning a streak that's made odds makers call him the "most dangerous sports bettor in Nevada." Lara Logan reports.

Sports Bettor Billy Walter's Winning Streak

The Super Bowl: America's Biggest Bet

### Safe prime

A safe prime is a prime number of the form 2p + 1, where p is also a prime

### Zuckerman number

In base 10, the first few Zuckerman numbers with more than one digit are
11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384,...

One example of a 4-digit number is : 1115

And 1115 is the smallest four-digit integer, abcd, such that abcd/(a*b*c*d) is a prime number.

### 4-digit numbers ending with 1

I'm trying to find something interesting to discuss about such numbers:

1101
1201 = 24^2 + 25^2 ........... 1201^2 = 49^2 + 1200^2
1301 = 25^2 + 26^2 ........... 1301^2 = 51^2 + 1300^2
1401
1501
1601
1701
1801
1901
2001
2101
2201
2301
2401
2501
2601
2701
2801
2901

etc

3-digit numbers
-----------------------
811
821
831
841
851
861
871
881 = 16^2 + 25^2 .............. 881^2 = 369^2 + 800^2
891
901
911
921
931
941
951

### Inserting a zero between adjacent digits of a number to produce a prime number

There are many examples, one such number is : 553

The factors of 553 are : 1 7 79 553
The prime factors are: 7 * 79

The factors of 5053 are : 1 31 163 5053
The prime factors are: 31 * 163

The factors of 5503 are : 1 5503
The prime factors are: 5503 is a prime number.

The factors of 50503 are : 1 50503
The prime factors are: 50503 is a prime number.

### 7^(7^n) + 1

Prove that for every nonnegative integer n, the number

7^(7^n) + 1

is the product of at least 2n + 3 (not necessarily distinct) primes.

## Thursday, January 20, 2011

### 2^p + 3^p = a^b

a and b are positive integers, and p is a prime number that satisfy this equation:

2^p + 3^p = a^b

Can b > 1?

### Leyland number

A Leyland number is a number of the form x^y + y^x,
where x and y are integers greater than 1

Primes of form x^y + y^x

### New theories reveal the nature of numbers

http://esciencecommons.blogspot.com/2011/01/new-theories-reveal-nature-of-numbers.html

Ono and his research team have discovered that partition numbers behave like fractals. They have unlocked the divisibility properties of partitions, and developed a mathematical theory for “seeing” their infinitely repeating superstructure. And they have devised the first finite formula to calculate the partitions of any number.

Hidden Structure to Partition Function

Mathematicians find a surprising fractal structure in number theory

## Wednesday, January 19, 2011

### Playing with Factorials

We take 4 consecutive integers such that

((1 !) + (2 !) + (3 !) + (4 !)) / (1 * 2 * 3 * 4) = 1.37500

((2 !) + (3 !) + (4 !) + (5 !)) / (2 * 3 * 4 * 5) = 1.26666667

((3 !) + (4 !) + (5 !) + (6 !)) / (3 * 4 * 5 * 6) = 2.41666667

((4 !) + (5 !) + (6 !) + (7 !)) / (4 * 5 * 6 * 7) = 7.02857143

((5 !) + (6 !) + (7 !) + (8 !)) / (5 * 6 * 7 * 8) = 27.5

((6 !) + (7 !) + (8 !) + (9 !)) / (6 * 7 * 8 * 9) = 135.238095

((7 !) + (8 !) + (9 !) + (10 !)) / (7 * 8 * 9 * 10) = 801

We get our first integer

If we continue onto the next series, we get

((8 !) + (9 !) + (10 !) + (11 !)) / (8 * 9 * 10 * 11) = 5549.09091

This is not an integer. Could you predict when we could get an integer?

### Square numbers using the same digits

For example,

245^2 = 60025,
160^2 = 25600,
250^2 = 62500

Could you find other examples?

### Sums of Powers

A collection of sums of 2 squares, sums of 2 cubes, sums of 3 squares, sums of 3 cubes, and of other powers.

1 = 3^2 - 2^3
2 = 3^3 - 5^2
2 = 4^2 - 3^2 - 2^2 - 1^2
3 = 2^2 - 2^0
3^2 = 5^2 - 4^2
3^3 = 6^3 - 5^3 - 4^3
3^3 = 3^2 + 3^2 + 3^2
4 = 3^2 - 2^2 - 1^2
5 = 3^2 - 2^2 = 1^2 + 2^2
5 = 2^5 - 3^3 ............ 5^2 = 3^2 + 4^2 = 13^2 - 12^2
6 = 3 + 2 + 1 .......... (the smallest perfect number)
7 = 2^5 - 5^2 ................... (7 is quite remarkable, notice the digits)
7 = 4 + 3 = 4^2 - 3^2
7 = 1^2 + 1^2 + 1^2 + 2^2
8 = 2^4 - 2^3 ........ = 2^2 + 2^2
9 = 1^3 + 2^3
10 = 2 + 3 + 5 (sum of first prime numbers) .... 100 = 10^2 = (2 + 3 + 5)^2
11 = 6 + 5 = 6^2 - 5^2
12 = 3^1 + 3^2
12 = 2^4 - 2^2
12 = 4^2 - 4^1
13 = 2^2 + 3^2 = 7^2 - 6^2 .... 13^2 = 8^3 - 7^3
13 = 2^2 + 3^2 ........... 13^2 = 5^2 + 12^2 ... 13^4 = 120^2 + 121^2
14 = 1^2 + 2^2 + 3^2
14 = 1 + 4 + 9 = 1^2 + 2^2 + 3^2
14 = 2 + 3 + 4 + 5 (sum of consecutive integers)
14 = 2^1 + 2^2 + 2^3
15 is a triangular number: 15 = 1 + 2 + 3 + 4 + 5
15 = 8^2 - 7^2
15 = 4^2 - 1^2
16 = 1 + 3 + 5 + 7 (sum of the four first odd numbers)
17 = 2 + 3 + 5 + 7 (sum of 4 consecutive prime numbers)
17 = 2^3 + 3^2 = 1^3 + 4^2 .................. = 1^4 + 2^4
17 = = 9^2 - 8^2 = 3^4 - 4^3 ........ 17^2 = 1^3 + 2^3 + 4^3 + 6^3
17 = 2^(2^2) + 1 (third Fermat prime)
17 = 2 + 2 + 13 = 3 + 3 + 11 = 3 + 7 + 7 = 5 + 5 + 7
(17 is the smallest number with 4 representations as a sum of 3 primes)
18 = 3^3 - 3^2
18 = 3 + 4 + 5 + 6 (sum of consecutive numbers)
19 = 3^3 - 2^3 (difference of consecutive cubes)
20 = 6^2 - 4^2 ........... 20^2 = 7^0 + 7^1 + 7^2 + 7^3 = 29^2 - 21^2
20 = 1 + 3 + 6 + 10 (sum of the first 4 triangular numbers)
20 = 1 + 1 + 2 + 3 + 5 + 8 (sum of the first 6 Fibonacci numbers)
20 = 2 + 4 + 6 + 8 (sum of the first 4 even numbers)
21 = 4^0 + 4^1 + 4^2
22 = 4 + 5 + 6 + 7
22 = 1^4 + 2^3 + 3^2 + 4^1
23 = 5 + 7 + 11 (smallest prime number that is a sum of 3 consecutive prime numbers)
23 = 12^2 - 11^2
23 = 1^4 + 2^3 + 3^2 + 4^1 + 5^0
24 = 3 + 5 + 7 + 9 (sum of consecutive odd numbers)
24 = 2^5 - 2^3
24 = 3^3 - 3^1
25 = 1^2 + 2^2 + 2^2 + 4^2
25 = 4^2 + 3^2 ........... 25^2 = 7^2 + 24^2
26 = 5 + 8 + 13 (sum of consecutive Fibonacci numbers)
27 (=3^3) = 3^2 + 3^2 + 3^2
27 = 1^2 + 1^2 + 5^2 ............... 19683 = 27^3 = 3^3 + 18^3 + 24^3
27 = 3^2 + 3^2 + 3^2
27 = 6^2 - 3^2
27 = 14^2 - 13^2 (difference of consecutive squares)
27 = 2 + 3 + 4 + 5 + 6 + 7 (sum of consecutive natural numbers)
28 = 1 + 5 + 9 + 13 (hexagonal number)
28 = 2 + 3 + 5 + 7 + 11 (sum of the first five consecutive primes)
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 (sum of the first 7 consecutive numbers)
28 = 2^5 – 2^2
29 is the smallest prime of the form 7n + 1 ... 29^2 = 21^2 + 20^2 (Pythagorean triple)
29 = 2^2 + 3^2 + 4^2 (sum of 3 consecutive squares)
29 = 3 + 5 + 7 + 11 + 13 (sum of consecutive primes)
29 = 2^2 + 3^2 + 4^2
30 = 1^2 + 2^2 + 3^2 + 4^2
30 = 9 + 10 + 11 = 6 + 7 + 8 + 9 = 4 + 5 + 6 + 7 + 8
31 = 2^2 + 3^3 ............................... ..... = 2^5 - 1
31 = 5^0 + 5^1 + 5^2
31 = 2^0 + 2^1 + 2^2 + 2^3 + 2^4
32:
33:
34:
35:
36 = 2^2 + 4^2 + 4^2
36 (=6^2) = 1^3 + 2^3 + 3^3
36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19
(smallest number with four representations as a sum of two distinct prime numbers)
37:
38:
39:
40:
41:
42:
43:
44:
45:
46:
47:
48 = 4^2 + 4^2 + 4^2
48 = 3 + 5 + 7 + 9 + 11 + 13
48 = 5 + 43 = 7 + 41 = 11 + 37 = 17 + 31 = 19 + 29
(the smallest number with five representations as a sum of two primes)
49:
50 = 1^2 + 7^2 = 5^2 + 5^2 = 3^2 + 4^2 + 5^2
51 = 2^3 + 2^3 + 2^3 + 3^3
52 = 5^2 + 3^3
53 = 2^2 + 7^2 = 1^2 + 4^2 + 6^2 ......... 53^2 = 28^2 + 45^2
54 = 3^3 + 3^3
54 = 7^2 + 2^2 + 1^2 = 6^2 + 3^2 + 3^2 = 5^2 + 5^2 + 2^2
(54 is the smallest number that can be written as the sum of 3 squares in 3 different ways)
55 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2
56:
57 = 1^2 + 2^2 + 4^2 + 6^2 ................................. = 7 + 8 + 9 + 10 + 11 + 12
58 = 3^2 + 7^2 ....... = 2^2 + 2^2 + 5^2 + 5^2
59 = 2^3 + 2^3 + 2^3 + 2^3 + 3^3
60:
61 = 5^2 + 6^2 ............................ 61^2 = 11^2 + 60^2
62 = 2^3 + 3^3 + 3^3
62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2
(62 is the smallest number that can be written as the sum of 3 distinct squares in 2 distinct ways)
63 = 6^2 + 3^3
64:
65 = 1^2 + 8^2 = 4^2 + 7^2 ........... = 1^5 + 2^4 + 3^3 + 4^2 + 5^1
66 = 2^2 + 2^2 + 3^2 + 7^2
67:
68:
69:
70 = 3^2 + 5^2 + 6^2 ........... 70^2 = 1^2 + 2^2 + 3^2 + 4^2 + ... + 22^2 + 23^2 + 24^2
71 -> 71^2 = 2^7 + 17^3 (sum of prime powers of two prime numbers)
71 -> 71^4 = 136^3 + 4785^2
72 = 2^3 + 4^3 ..... 72^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
73 = 1^3 + 2^3 + 4^3
73 = 3^2 + 8^2 ........................... 73^2 = 48^2 + 55^2 (Pythagorean triple)
74:
75 -> 75^5 = 19^5 + 43^5 + 46^5 + 47^5 + 67^5
76 = 2^2 + 6^2 + 6^2 .............................. = 3^2 + 3^2 + 3^2 + 7^2
77 = 4^2 + 5^2 + 6^2
78 = 2^2 + 5^2 + 7^2 ............ 78^3 = 39^3 + 52^3 + 65^3
79 = 2^2 + 5^2 + 5^2 + 5^2
80 = 4^2 + 8^2 = 4^2 + 4^3 ........ 80^2 = 4^3 + 8^3 + 12^3 + 16^3
80 = 2^3 + 2^3 + 4^3
81 = 1^2 + 4^2 + 8^2 ......... 2^5 + 7^2
82 = 1^2 + 9^2 .............. 2^2 + 2^2 + 5^2 + 7^2
83:
84 = 4^1 + 4^2 + 4^3
84 = 3^2 + 5^2 + 5^2 + 5^2
84 = 5 + 79 = 11 + 73 = 13 + 71 = 17 + 67 = 23 + 61 = 31 + 53 = 37 + 47 = 41 + 43
(smallest number with eight representations as a sum of two primes)
84 = 2^2 + 4^2 + 8^2
85 = 2^2 + 9^2 ........................... 85^2 = 13^2 + 84^2 = 36^2 + 77^2
85 = 6^2 + 7^2 .......... 4^0 + 4^1 + 4^2 + 4^3 (sum of powers of 4)
86 = 3^2 + 4^2 + 5^2 + 6^2 (sum of consecutive squares)
87 = 2^2 + 3^2 + 5^2 + 7^2
88:
89 = 8^1 + 9^2 ........... 89^2 = 39^2 + 80^2 (Pythagorean triple)
89 = 2^3 + 3^3 + 3^3 + 3^3
90 = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 (sum of consecutive squares)
90 = 9^1 + 9^2 = 10^2 - 10^1
91 = 3^3 + 4^3
92:
93 = 2^2 + 5^2 + 8^2
94:
95:
96 = 4^2 + 4^2 + 8^2
97:
98:
99 = 2^3 + 3^3 + 4^3
100 = 6^2 + 8^2 ....... = 1^3 + 2^3 + 3^3 + 4^3
101
102
103
104
105
106
107
108
109 = 3^2 + 10^2 ........... 109^2 = 60^2 + 91^2
110:
111
112
113
114
115 = 3^2 + 5^2 + 9^2

120
121
122
123
124
125
126
127
128 = 8^2 + 8^2
129
130:

137 = 4^2 + 11^2
138
139
140 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 (sum of the first seven squares)
140 = 2^2 + 6^2 + 10^2

150:

160:

168 = 2^2 + 8^2 + 10^2
169
170:

180:
181 = 9^2 + 10^2 ......................................................... 181^2 = 19^2 + 180^2

190:
191
192
193
194 = 5^2 + 13^2
194 = 1^2 + 7^2 + 12^2 = 3^2 + 4^2 + 13^2 = 3^2 + 8^2 + 11^2 = 5^2 + 5^2 + 12^2 = 7^2 + 8^2 + 9^2

200
201
202 = 9^2 + 11^2

216 (=6^3) = 3^3 + 4^3 + 5^3

242 = 11^2 + 11^2
243
244
245 = 7^2 + 14^2

258 = 59 + 61 + 67 + 71 (sum of four consecutive prime numbers)

269 = 10^2 + 13^2 ............... 269^2 = 69^2 + 260^2
270
271 = 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43

300:

320 = 8^2 + 16^2

330

340
341
342
343 = 7^3 = 18^0 + 18^1 + 18^2
344
345

369 = 12^2 + 15^2

352 = 8^2 + 12^2 + 12^2

385 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2

390
391
392
393
394
395

400

450 = 3^2 + 21^2 ............ = 223 + 227 (sum of two consecutive prime numbers)
451
452
453
454
455
456 = 107 + 109 + 113 + 127 (sum of four consecutive prime numbers)
457 = 149 + 151 + 157 (sum of three consecutive prime numbers)
458
459
460
461 = 444 + 6 + 11
462 = 67 + 71 + 73 + 79 + 83 + 89 (sum of six consecutive prime numbers)
463 = 53 + 59 + 61 + 67 + 71 + 73 + 79 (sum of seven consecutive prime numbers)
464
465
466
467
468 = 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67
(sum of ten consecutive prime numbers)
469
470

480
481 = 15^2 + 16^2 (sum of 2 consecutive integers)
482
483
484
485
486 = 3^5 + 3^5
487
488
489 is an octahedral number.
490 = 7^2 + 21^2 ........... = 1^5 + 1^5 + 1^5 + 1^5 + 3^5 + 3^5
491
492
493
494
495
496 is the third perfect number
496 = 1^3 + 3^3 + 5^3 + 7^3
496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 (A triangular number)
497 = 89 + 97 + 101 + 103 + 107 (sum of consecutive primes)
498
499 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 (sum of consecutive primes)
500
501
502
503
504
505
506 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2

520 = = 6^2 + 22^2 = 14^2 + 18^2 .................... = 257 + 263 (sum of two successive prime numbers)
521
522 = 9^2 + 21^2 ............... = 1^5 + 1^5 + 1^5 + 1^5 + 2^5 + 3^5 + 3^5
523
524
525

548 = 8^2 + 22^2
549
550
551
552
553 = 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79
554
555
556
567
568
569
560 = 4^2 + 12^2 + 20^2

570

580 = 2^2 + 24^2 = 16^2 + 18^2
581 = 191 + 193 + 197

600:
601
602
603
605
606
607
608
609
610
611
612
613
614
615 = 9^2 + 10^2 + 11^2 + 12^2 + 13^2 (sum of five consecutive squares)
616
617
618
619
620
621
622
623
624
625
626 = 1^2 + 25^2 .... 626 = 5^4 + 1^4
627
628
629
630

640

650

660
661
662
663
664
665
666 = 3^6 - 2^6 + 1^6
666 = 6^3 + 6^3 + 6^3 + 6 + 6 + 6
666 = 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2

670
671
672 = 4^2 + 16^2 + 20^2
673
674
675

700:

713 = 233 + 239 + 241

725 = 7^2 + 26^2 = 10^2 + 25^2 = 14^2 + 23^2 ... 725^2 = 333^2 + 644^2 = 364^2 + 627^2

736 = 4^2 + 12^2 + 24^2
737
738
739
740

750:

776 = 10^2 + 26^2

799 = 3^5 + 3^5 + 3^5 + 2^5 + 2^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5 + 1^5
800:
801 = 15^2 + 24^2
801 = (7! + 8! + 9! + 10!)/(7 * 8 * 9 * 10)

810
811 = 151 + 157 + 163 + 167 + 173
(sum of five consecutive primes)
811 (prime of the form 4n + 3) = 405 + 406
812
813
814
815

820 = 6^2 + 28^2 = 12^2 + 26^2

829 = 10^2 + 27^2 .............. 829^2 = 540^2 + 629^2

842 = 1^2 + 29^2

850:

859 = 1^5 + 1^5 + 2^5 + 2^5 + 2^5 + 2^5 + 3^5 + 3^5 + 3^5
860
861
862
863
864
865
866 = 5^2 + 29^2

881 = 16^2 + 25^2 .................................... 881^2 = 369^2 + 800^2

900
901 = 1^2 + 30^2 = 15^2 + 26^2
902
903
904
905
906
907 = 3^2 + 13^2 + 27^2

917 = 173 + 179 + 181 + 191 + 193
918
919
920
921
922
923
924
925
926
927
928
929 = 20^2 + 23^2 .... 929^2 = 129^2 + 920^2 .... 929^3 = 69^3 + 447^3 + 893^3
930

945 = 472 + 473
945 = 314 + 315 + 316
945 = 187 + 188 + 189 + 190 + 191
945 = 155+156+157+158+159+160
945 = 132+133+134+135+136+137+138
945 = 101+102+103+104+105+106+107+108+109
945 = 90+91+92+93+94+95+96+97+98+99
945 = 61+62+63+64+65+66+67+68+69+70+71+72+73+74
945 = 56+57+58+59+60+61+62+63+64+65+66+67+68+69+70
945 = 44+45+46+47+48+49+50+51+52+53+54+55+56+57+58+59+60+61
945 = 35+36+37+38+39+40+41+42+43+44+45+46+47+48+49+50+51+52+53+54+55
945 =22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46+47+48
945 = 17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44+45+46
945 = 10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43+44
945 = 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43

949 = 7^2 + 30^2 = 18^2 + 25^2

966 = 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139
(sum of eight consecutive primes)

980
981
982
983
984 = 8 + 88 + 888
985
986
987
988
989
990
991
992 = 8 + 8 + 88 + 888
993
994
995
996
997
998
999
1000 = 8 + 8 + 8 + 88 + 888

1115 = 103 + 107 + 109 + 113 + 127 + 131 + 139 + 149

1201 = 24^2 + 25^2 ............................. 1201^2 = 49^2 + 1200^2

1233 = 12^2 + 33^2
1233 is the smallest 2n-digit number that is the sum of the squares of its n-digit halves

1301 = 25^2 + 26^2 ............................. 1301^2 = 51^2 + 1300^2

1412 = 16^2 + 34^2

1717 = 6^2 + 41^2 = 14^2 + 39^2 ..... 1717^2 = 492^2 + 1645^2 = 1092^2 + 1325^2

1900 = 223 + 227 + 229 + 233 + 239 + 241 + 251 + 257

1927 = 2^11 – 11^2

1949 = 10^2 + 43^2 ............................................ 1949^2 = 860^2 + 1749^2

2000 = 8^2 + 44^2 = 20^2 + 40^2
2001
2002
2003
2004
2005
2006
2007
2008
2009 = 28^2 + 35^2
2010

2100

2200

2300

2357 = 26^2 + 41^2 ................................................ 2357^2 = 1005^2 + 2132^2
2357 = 773 + 787 + 797 ............................. 2357 = 461 + 463 + 467 + 479 + 487

2400
2401 = 7^4

2500
2501 = 1^2 + 50^2 = 10^2 + 49^2 ......... 2501^2 = 100^2 + 2499^2 = 980^2 + 2301^2

3000

3025 = 55^2 = 44^2 + 33^2

4000

4253 = 38^2 + 53^2 ............................................ 4253^2 = 1365^2 + 4028^2

5000

5008 = 48^2 + 52^2

5557 = 9^2 + 74^2 ..... 5557^2 = 1332^2 + 5395^2
2 + 3 + 5 + 7 + 11 + . . . + 3833 = 3847 + 3851 + . . . + 5557

## Tuesday, January 18, 2011

### Computer-assisted proof

http://en.wikipedia.org/wiki/Computer-assisted_proof

Some mathematicians believe computer-assisted proofs are not 'real' mathematical proofs.

### A couple of records in number theory

by Antoine Joux
Presented at the Crypto '98 rump session.

### Number Theory - Videos

Introduction to Number Theory 01

View more lectures at

### Convolution Formula

MIT Math Lecture: Differential Equations - 21 - Convolution Formula

Lecture 21: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems. Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.

These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring 2003 and do not correspond precisely to the lectures taught in the Spring of 2006. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectures. The videotaping was made possible by The d'Arbeloff Fund for Excellence in MIT Education. Note: Lecture 18, 34, and 35 are not available. This material was created or adapted from material created by MIT faculty member Arthur Mattuck, Professor. Copyright © 2003 Arthur Mattuck. For more videos from MIT, please visit: ocw.mit.edu. For more education videos, please visit: www.CosmoLearning.com

### Some Problems from Diophantus’s Arithmetika

http://www.math.uconn.edu/~leibowitz/math2720f08/diophantusexamples.pdf

Diophantus of Alexandria

One of his propositions states that the difference of the cubes of two numbers can always be expressed as the sum of the cubes of two other numbers; that no number of the form 4n – 1 can be expressed as the sum of two squares; and that no number of the form 8n – 1 can be expressed as the sum of three squares

Given two cubes, to find in rational numbers two others such that their difference is equal to the sum of the given cubes.

That is to say, solving a^3 + b^3 = x^2 - y^2

x = a(a^3 + 2b^3) / (a^3 - b^3)
y = b(2a^3 + b^3) / (a^3 - b^3)

Given two cubes, to find in rational numbers two cubes such that their difference is equal to the difference of the given cubes.

That is to say, a^3 - b^3 = x^3 - y^3

Vieta finds

x = b(2a^3 - b^3) / (a^3 + b^3)
y = a(2b^3 - a^3) / (a^3 + b^3)

### Sum and Difference of Two Cubes

http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.pdf

Differences between two positive cubes in exactly 1 way

Differences between two positive cubes in exactly 2 ways

Differences between two positive cubes in exactly 3 ways

Difference between two positive cubes in more than one way

Differences between positive cubes in 1, 2 or 3 ways:
union of A014439, A014440 and A014441

Numbers n such that n and n+1 are differences between 2 positive cubes
in at least one way

Numbers n such that n and n-1 are differences between 2 positive cubes
in at least one way

Numbers n such that n is a perfect square and is a difference
between 2 positive cubes in at least one way

Numbers n such that n^2 is a difference between 2 positive cubes
in at least one way

Differences between numbers that are a difference between 2 positive cubes
in at least one way

Numbers n such that n-th and (n+1)st term of A038593 differ by 1

Numbers that are not the difference between two positive cubes

Palindromic numbers which are the difference of two positive cubes

Odd numbers that are differences between two cubes in at least one way

Even numbers that are differences between two cubes in at least one way

Numbers that are divisible by 4 and are differences between two cubes
in at least one way

Numbers that are divisible by 8 and are differences between two cubes
in at least one way

Divisible by 3 (and 9) and are differences between two cubes in at least one way

Numbers that are divisible by 6 (and 18) and are differences between two cubes
in at least one way

Numbers that are divisible by 5 and are the difference between two
(different positive) cubes in at least one way

Numbers that are divisible by 10 and are differences between two cubes
in at least one way

Numbers that are divisible by 7 and are differences between two cubes
in at least one way

Numbers n such that n ends with '1' and is difference between two cubes
in at least one way

Numbers n such that n ends with '2' and is difference between two cubes
in at least one way

Numbers n such that n ends with '3' and is difference between two cubes
in at least one way

Numbers n such that n ends with '4' and is difference between two cubes
in at least one way

Numbers that end in '5' and are the difference between two (positive) cubes
in at least one way

Numbers n such that n ends with '6' and is difference between two cubes
in at least one way

Numbers n such that n ends with '7' and is the difference between two cubes
in at least one way

Numbers n such that n ends with '8' and is difference between two cubes
in at least one way

Numbers n such that n ends with '9' and is difference between two cubes
in at least one way

Numbers whose square is expressible as the difference of positive cubes
in more than one way

Semiprimes that can be expressed as the sum or difference of two cubes.
Common terms of A001358 and A045980

a(n) = 15n^2 + 13n^3

Positive sums or differences of two cubes of primes

Smallest number that is the difference between two positive cubes in n ways

Numbers expressible as sum or difference of two cubes of primes
in at least two ways

Triangular numbers which are differences of nonnegative cubes

(n) = number of solutions to x^3 - y^3 == 0 (mod n)

Numbers of the form x^3 + y^3 or x^3 - y^3
Product of three solutions of the Diophantine equation x^3 - y^3 = z^2
Numbers expressible as the difference of two nonnegative cubes

Sequence and first differences (A030124) include all numbers exactly once

Complement (and also first differences) of Hofstadter's sequence A005228

## Monday, January 17, 2011

### STEM Education

STS-134 Mission Information

NASA's Shuttle and Rocket Launch Schedule

SSEP is Proud to Announce the Student Proposals Selected for Spaceflight
on STS-134

## Saturday, January 15, 2011

### Prime p as the sum of two squares

Which primes p can be written as the sum of two squares,
and in how many ways can it be done?

### Maths Humor

Maths Integral Joke

Ma and Pa Kettle - Math Lesson

Abbott And Costello 13 X 7 is 28 from the movie A Little Giant

## Friday, January 14, 2011

### APR vs Interest Rate

APR Calculator

Effective interest rate

### David Mumford: Math marching to a different drummer

http://www.dam.brown.edu/people/mumford/

A distinguished mathematician and highly recognized scholar, Dr. David Mumford is a leader in the field of pattern theory. He began his career in pure mathematics, specifically in studying the moduli spaces of curves and algebraic varieties. In the past two decades, Mumford has collaborated with computer scientists, psychologists, and neurobiologists and sought the “right” mathematics for describing the problems of perception. His work has focused on computer vision, pattern theory, and the mathematical modeling of shape. His most recent publications include Indra’s Pearls: The Vision of Felix Klein (Cambridge University Press, 2002) and Selected Papers on the Classification of Varieties and Moduli Spaces (Springer-Verlag, 2004).

## Thursday, January 13, 2011

### Websites and Blogs

Vi Hart Vi Deos

It covers many areas (quantum computing, perception, cognitive science, artificial intelligence, evolutionary biology, theoretical physics, theoretical mathematics, etc).

http://www.jeffreyepsteinscience.com/

### The On-Line Encyclopedia of Integer Sequences

There is a new version of the On-Line Encyclopedia of Integer Sequences (the OEIS). There are two parts: the main OEIS page, which has all the sequence data, and a wiki, which has discussion pages.

### A modified Turing Test Machine

Passing The Turing Test
Can you tell a human from a machine?

Alan Turing and the Imitation Game

Problem :

Suppose you have a modified Turing machine that can predict the future -- it can only give you information about a week ahead of time.

So you want to test the machine, and ask the machine to give you the lottery numbers for the next drawing of the 6/49 lottery. The machine cannot give you 6 numbers at the same time. It cannot handle this.

So you ask for the sum of the 6 lottery numbers.
The machine gives you that number.

However, there's a very large number of possible combinations that yield this sum.

Then, the machine tells you: If figure out exactly how many possible number combinations there are (from 6 out of 49 numbers) to yield this particular sum, and then multiply this number with the sum it will yield a new number of around one million, and you will also get the same number if you multiply all 6 lottery numbers with each other.

The ball is in your court. The machine awaits.

You need to figure out those 6 numbers. Solve this problem.

### Fantasy football, plus math, equals motivated students

How do fantasy football and math class coexist? How does it work?

http://www.necn.com/Boston/SciTech/2008/11/25/Fantasy-football-plus-math/1227664512.html

## Wednesday, January 12, 2011

### Old posts : October 2010

Friday, October 15, 2010
3 lucky dates: 8-8-8, 9-9-9, 10-10-10

http://benvitale-3luckydates.blogspot.com/2010/10/3-lucky-dates-8-8-8-9-9-9-10-10-10.html?spref=bl

Integers that are both perfect squares and cubes

Waring's problem for cubes

### Patterns with division

1/7 = 0.142857142857142857...

Take 142857, split it and then add the parts

142 + 857 = 999
14 + 28 + 57 = 99

2/7 = 0.285714285714285714....

285714 :

285 + 714 = 999
28 + 57 + 14 = 99

3/7 = 0.428571428571428571...

428571 :

428 + 571 = 999
42 + 85 + 71 = 198 = 2 * 99

4/7 = 0.571428571428571428....

571428

571 + 428 = 999
57 + 14 + 28 = 99

5/7 = 0.714285714285714285...

714285 :

714 + 285 = 999
71 + 42 + 85 = 198 = 2 * 99

6/7 = 0.857142857142857142...

857142 :

857 + 142 = 999
85 + 71 + 42 = 198 = 2 * 99

1/13 = 0.076923076923076923....
076 + 923 = 999
07 + 69 + 23 = 99

2/13 = 0.153846153846153846....
153 + 846 = 999
15 + 38 + 46 = 99

3/13 = 0.230769230769230769....
230 + 769 = 999
23 + 07 + 69 = 99

4/13 = 0.307692307692307692....
307 + 692 = 999
30+ 76 + 92 = 198 = 2 * 99

5/13 = 0.384615384615384615....
384 + 615 = 999
38 + 46 + 15 = 99

6/13 = 0.461538461538461538....
461 + 538 = 999
46 + 15 + 38 = 99

7/13 = 0.538461538461538461....
538 + 461 = 999
53 + 84 + 61 = 198 = 2 * 99

8/13 = 0.615384615384615384....
615 + 384 = 999
61 + 53 + 84 = 198 = 2 * 99

9/13 = 0.692307692307692307....
692 + 307 = 999
69 + 23 + 07 = 99

10/13 = 0.769230769230769230....
769 + 230 = 999
76 + 92 + 30 = 198 = 2 * 99

11/13 = 0.846153846153846153....
846 + 153 = 999
84 + 61 + 53 = 198 = 2 * 99

12/13 = 0.923076923076923076....
923 + 076 = 999
92 + 30 + 76 = 198 = 2 * 99

To Be Continued

## Tuesday, January 11, 2011

### Repunits

Repunits: (10^n - 1)/9. Often denoted by R_n.

0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111

http://oeis.org/A002275

If p(1) = 1, p(2) = 11, p(3) = 111, p(4) = 1111, etc.

where p(n) a 10-base integer represented by a string of n ones.

Most of the repunit numbers are composite.

Indices of prime repunits: numbers n such that 11...111 = (10^n - 1)/9 is prime.

2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343

How do you prove that for p(n) to be prime n has to be a prime number?