The Hidden Mathematical Beauties in Number Relations – Part 1

Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. . . . A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

George PÓLYA

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We are all aware of various relations, such as family relations, work relations, and so on. While dealing with numbers, we can find several hidden relations among them using various mathematical operations. For example, 48 + 17 = 50 + 15, 23 ⎼ 9 = 24 ⎼ 10, √81 = 8 + 1, and so on. In this blog post, we will consider one such number relation.

Consider the numbers 123789, 561945, 642864, 242868, 761943 and 323787. They are all six-digit numbers. Is there any connection between them? Certainly it’s challenging, but we have the following relation:

123789² + 561945² + 642864² ‒ 242868² ‒ 761943² ‒ 323787² = 0.

Let us call this the original equation. A direct computation shows that

123789² + 561945² + 642864² ‒ 242868² ‒ 761943² ‒ 323787²

= 15323716521 + 315782183025 + 413274122496 ‒ 58984865424 ‒ 580557135249 ‒ 104838021369

= 744380022042 ‒ 744380022042

= 0.

Certainly, this is interesting. While we are seeing the numbers for the first time, it’s not so easy to figure out the existence of such a relationship. Even more interesting is that, if we delete the extreme left digits from each of these six numbers (and the newly formed numbers are 23789, 61945, 42864, 42868, 61943 and 23787), they are still related by the following relation:

23789² + 61945² + 42864² ‒ 42868² ‒ 61943² ‒ 23787² = 0.

A simple computation shows that

23789² + 61945² + 42864² ‒ 42868² ‒ 61943² ‒ 23787²

= 565916521 + 3837183025 + 1837322496 ‒ 1837665424 ‒ 3836935249 ‒ 565821369

= 6240422042 ⎼ 6240422042

= 0.

Now, that’s not all. More is on the way. In fact, if we repeat the above procedure (deleting the extreme left digits from each of these six numbers to form the new numbers 3789, 1945, 2864, 2868, 1943 and 3787) one more time, the relation still holds good. Thus, we have the following relation:

3789² + 1945² + 2864² ‒ 2868² ‒ 1943² ‒ 3787² = 0.

A direct computation shows that

3789² + 1945² + 2864² ‒ 2868² ‒ 1943² ‒ 3787²

= 14356521 + 3783025 + 8202496 ‒ 8225424 ‒ 3775249 ‒ 14341369

= 26342042 ⎼ 26342042

= 0.

Following a similar procedure as above, we have

789² + 945² + 864² ‒ 868² ‒ 943² ‒ 787²

= 622521 + 893025 + 746496 ‒ 753424 ‒ 889249 ‒ 619369

= 2262042 ⎼ 2262042

= 0.

Deleting the leftmost digits once again, and following the same footprints once again, we get

89² + 45² + 64² ‒ 68² ‒ 43² ‒ 87²

= 7921 + 2025 + 4096 – 4624 – 1849 – 7569

= 14042 ⎼ 14042

= 0.

Finally deleting the leftmost digits for the last time, we have

9² + 5² + 4² ‒ 8² ‒ 3² ‒ 7²

= 81 + 25 + 16 – 64 – 9 – 49

= 122 ‒ 122

= 0.

But that’s not all; more to come out of the original equation! Instead of successively deleting the extreme leftmost digits from the original equation, if we delete the rightmost digits in succession, the resulting relations are also true. Let us examine the relationship following the first truncation, which is

12378² + 56194² + 64286² ‒ 24286² ‒ 76194² ‒ 32378² = 0.

We have

12378² + 56194² + 64286² ‒ 24286² ‒ 76194² ‒ 32378²

= 15323716521 + 315782183025 + 413274122496 ‒ 58984865424 ‒ 580557135249 ‒ 10483802136

= 744380022042 ⎼ 744380022042

= 0.

Once again,

1237² + 5619² + 6428² ‒ 2428² ‒ 7619² ‒ 3237²

= 1530169 + 31573161 + 41319184 ‒ 5895184 ‒ 58049161 ‒ 10478169

= 74422514 ⎼ 74422514

= 0.

Further, we get

123² + 561² + 642² ‒ 242² ‒ 761² ‒ 323²

= 15129 + 314721 + 412164 ‒ 58564 ‒ 579121 ‒ 104329

= 742014 ⎼ 742 014

= 0.

Also, we have

12² + 56² + 64² ‒ 24² ‒ 76² ‒ 32²

= 144 + 3136 + 4096 ‒ 576 ‒ 5776 ‒ 1024

= 7376 ⎼ 7376

= 0.

Finally, we have

1² + 5² + 6² ‒ 2² ‒ 7² ‒ 3²

= 1 + 25 + 36 ‒ 4 ‒ 49 ‒ 9

= 62 ‒ 62

= 0.

Is that all? Can we find any more hidden beauties in the original equation? Certainly yes, and it is a mixture of the previous two! In other words, if we start deleting the extreme left and right digits together from the original equation, we can see the following hidden beauties in the original equation:

2378² + 6194² + 4286² ‒ 4286² ‒ 6194² ‒ 2378² = 0.

In fact, the above relation is quite obvious.

Repeating the procedure one more time, we have 37² + 19² + 28² ‒ 28² ‒ 19² ‒ 37² = 0, which is also obvious.

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