What makes the numbers 142[999]857 special?

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Discussion Overview

The discussion centers around the properties and significance of the number 142[999]857, exploring its mathematical characteristics and relationships, particularly in relation to its decimal representation and multiplicative patterns. Participants examine various aspects, including divisibility, repeating sequences, and numerical relationships.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants note that 1/7 results in a repeating decimal of 0.142857, suggesting a connection to the number 142[999]857.
  • One participant claims that 142 + 857 equals 999 and proposes that 142999857 is divisible by 142, 999, and 857, although this has not been verified.
  • Another participant challenges the relevance of the sum 142 + 857 = 999 to the discussion.
  • Several participants discuss the properties of the digits in 142857, including various arithmetic operations that yield specific results, but the significance of these operations remains unclear.
  • A participant argues that while 142857 is a repeating sequence, it may not be particularly special compared to other repeating decimals like those derived from 1/17.
  • One participant presents a pattern in the products of 142[999]857, noting that multiplying it by integers from 2 to 6 results in combinations of the same digits, regardless of the repeated nines.
  • Another participant expresses curiosity about the underlying reasons for the observed patterns in the products of 142[999]857.

Areas of Agreement / Disagreement

Participants express differing views on what makes the number 142[999]857 special, with some proposing various properties while others contest their significance. The discussion remains unresolved regarding a consensus on the number's uniqueness.

Contextual Notes

Some claims about divisibility and properties of the number are presented without verification, and the relevance of certain arithmetic operations is debated. The discussion includes multiple interpretations and lacks definitive conclusions.

Who May Find This Useful

Individuals interested in number theory, mathematical patterns, and properties of repeating decimals may find this discussion relevant.

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

1/7 = 0.14285714285714..
 
Last edited:
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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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