Blog Post 71: Number 343 in the spotlight!

343 is well known for its following characteristics as I had published in the earlier post

343 - 7 * 7 * 7
343 can be expressed as (3 + 4) ^ 3 by inserting few symbols between the digits
343 is one the few cubic or square palindrome

Taking this further let's look at the following series

Series 1: Insert 0 to 9 as the second digit (for eg. 3043, 3143, etc) and write down their factors

Series 2 : Prefix 0 to 9 before 343 and write down their factors

Are there any observations we could make or is it exhibited by most of the numbers?

Series 1                                                               Series 2

3043 – 17, 179                                                    0343 – 7,49

3143 – 7, 449                                                      1343 – 17, 79

               

3243 – 3, 23, 47                                                  2343 – 3, 11, 71

3343 – Prime                                                      3343 - Prime

3443 – 11, 313                                                    4343 – 43 and 101

3543 – 3, 1181                                                    5343 – 3, 13, 137

3643 – Prime                                                      6343 - Prime

3743 – 19, 197                                                    7343 – 7, 1049

3843 – 9, 61, 7                                                    8343 – 9, 9, 103

3943 – Prime number                                         9343 - Prime

Blog Post #70 - Sum of successive squares - what do we observe?

Take a look at the series below - first 20 set of sum of successive squares

1 ^ 2 + 2 ^ 2 = 5
2 ^ 2 + 3 ^ 2 = 13
3 ^ 2 + 4 ^ 2 = 25
4 ^ 2 + 5 ^ 2 = 41
5 ^ 2 + 6 ^ 2 = 61
6 ^ 2 + 7 ^ 2 = 85
7 ^ 2 + 8 ^ 2 = 113
8 ^ 2 + 9 ^ 2 = 145
9 ^ 2 + 10 ^ 2 = 181
10 ^ 2 + 11 ^ 2 = 221
11 ^ 2 + 12 ^ 2 = 265
12 ^ 2 + 13 ^ 2 = 313
13 ^ 2 + 14 ^ 2 = 365
14 ^ 2 + 15 ^ 2 = 421
15 ^ 2 + 16 ^ 2 = 481
16 ^ 2 + 17 ^ 2 = 545
17 ^ 2 + 18 ^ 2 = 613
18 ^ 2 + 19 ^ 2 = 685
19 ^ 2 + 20 ^ 2 = 761
20 ^ 2 + 21 ^ 2 = 841

What all do we observe and are there specific inferences?

Blog Post #69 - Is this pattern a unique behaviour or just a coincidence?

Consider these numbers

131 - Prime number, Its anagrams 113 and 311 are prime as well
919 - Prime number, Its anagrams 199 and 991 are prime as well
373 - Prime number, Its anagrams 733 and 337 are prime as well

Is there a pattern or does it break somewhere? Does it also extend to 4 digit numbers?

Even if it breaks somewhere is there something that is common to all these numbers?