What makes the numbers 142[999]857 special?

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The discussion centers around the unique properties of the number 142[999]857, particularly its relationship with the repeating decimal of 1/7, which is 0.142857. Participants explore various mathematical characteristics, noting that the sum of the digits (1+4-2+8-5+7) results in 13, while an alternate calculation yields -11, leading to a difference of 2. This ties back to the number 7, suggesting a deeper connection. The number is highlighted for its short repeating sequence and its divisibility properties, although some participants express skepticism about its significance. A key point is the multiplication of 142[999]857 by integers 2 through 6, which consistently produces combinations of the same digits, indicating a fascinating aspect of its structure. The thread encourages further exploration into number theory to understand these phenomena better.
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What makes the numbers 142[999]857 special?

1/7 = 0.14285714285714..
 
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Well, 142+857=999, and I would assume that 142999857 is divisible with 142,999 and 857 (I haven't checked the divisibility criterion though)[/color]
 
142+857=999 is not part of the intended solution
 
But the rest is?
 
Master of the Obvious

AntonVrba said:
What makes the numbers 142[999]857 special?

1/7 = 0.14285714285714..

:rolleyes: It only repeats as long as numbers exist?

:rolleyes: Individually they are all less than 10?

:rolleyes: It is a repeating sequence?

:rolleyes: 1+4-2+8-5+7 = 13 and 1-4+2-8+5-7 = -11 with 13-11 = 2 which is "14"/7 and "28"/7 = 4 which is twice 2 and "57" doesn't matter at all because it already has a "7" in it.

It is one of the only short sequence repeating numbers?
 
Rahmuss said:
:rolleyes: 1+4-2+8-5+7 = 13 and 1-4+2-8+5-7 = -11 with 13-11 = 2 which is "14"/7 and "28"/7 = 4 which is twice 2 and "57" doesn't matter at all because it already has a "7" in it.

It is one of the only short sequence repeating numbers?

nice try but none of these or others above

but I like the bit about repeating but what repeats?
 
There's nothing special about 142857.

For example:

1/17 = 0.0588235294117647 0588235294117647 0588235294117647 ...
2/17 = 0.1176470588235294 1176470588235294 1176470588235294 ...
3/17 = 0.1764705882352941 1764705882352941 1764705882352941 ...

Notice the cyclic nature of the decimal expansion.
 
AntonVrba said:
nice try but none of these or others above

but I like the bit about repeating but what repeats?

Well... the "142857" repeats; but that isn't really too special. I'm really either not sure what you're asking, or I'm stumped.
 
OK - her is the answer

2 x 142[999]857 = 285[999]714
3 x 142[999]857 = 428[999]571
upto
6 x 142[999]857= 857[999]124

the product 2x, 3x, 4x, 5x, 6x of 142[999]857 is always a combination of the same numbers, irrespective of how many times you repeat the nine in the middle
 
  • #10
Wow! That is kind of neat. Go into the number theory forum and see if they can give a good reason as to why that is. Or do you know why that is?
 

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