Blog #82 - The curious case of the numbers 7056 and 5776

I had posted earlier about some of the attributes of the number 7056 and 57767 and this is an extension of that

7056

a) 7056 is 84 x 84

b) Now check this sequence - 84, 70, 56 (This is the only 4 digit square which exhibits this sequence)

c) 70 ^ 2 - 56 ^ 2 = 1764 which is 42 x 42 and 42 is half of 84!

d) Also 42 extends the AP sequence highlighted in (b) = 84, 70, 56, 42!

e) Anagram of 42 is 24 and whose square is 576 (0576) which is anagram of 7056!

5776

a) 5776 is 76 x 76

b) Only 4 digit square (not ending with 25) whose last 2 digits are the original number itself

c) 57 - 19 x 3 and 76 - 19 x 4 and again one of the few squares which exhibit this sequence

d) Similar to 7056, 576 is again a subset of 5776 and a perfect square

e) Add 2000 to 5776 gives 7776 which is unique in its own sense - 3 ^ 5 x 2 ^ 5

Last but not the least - Prefix 1 to both 7056 and 5776

The resulting numbers are 17056 and 15776 and the similarities are striking

17056 - 2 ^ 5 x 13 x 41

15776 - 2 ^ 5 x 17 x 29

Both have 2 ^ 5 and a combination of 2 odd numbers as a factor

Post No. 81 - Ramanujam number and 2520!

This post is to showcase some of the interesting patterns generated when we combine two unique numbers

1729 - No introduction required... The Great Ramanujam number
2520 - Lowest number to have all factors from 1 to 10

Now for some interesting patterns

i) To start with 1729 + 2520 = 4249 which is 7 x 607.

ii) Now add 42 and 49 - we get 91 is a factor of 1729

iii) Add the digits of 607 - we get 13. Multiply by 7 and we get 91 again.

iv) Now add the digits of 4249 - we get 19!

v) 19 x 91 gives back 1729!

vi) Now add digits in pairs of 1729 and 2520 - 17 + 29 + 25 + 20 - Voila - it is 91 again!

vii) Now 2520 - 1729 = 791 which is 7 x 113 - Add the digits of 113 - we get 5. Multiply by 7 and we get 35. 35 is the odd factor component of 2520 (35 x 72)

We could go on creating patterns of this nature and there is no end at all but it is beautiful to see how some of the special numbers create further magic when they come together