Amazing Math Discovery: 8281 is the Only 4 Digit Square with this Quirk

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  • Thread starter Wilmer
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In summary, there is a unique 4-digit number, 8281, that follows the given conditions of having the first 2 digits be 1 greater than the last 2 digits and being a perfect square. However, this discovery unfortunately does not affect the price of groceries. The speaker then challenges someone to prove the uniqueness of this number without using a trial and error method. The conversation then continues with someone stating that they can prove it using intuition. The proof involves setting up an equation with the first and second digits represented by variables and concluding that there is only one solution, 8281.
  • #1
Wilmer
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Just discovered that:
if n is a 4 digit number such that the 2 digit number
formed by its 1st 2 digits is 1 greater than the
2 digit number formed by its last 2 digits,
and n is a square, then only one such n exists: 8281

Unfortunately, such a mathshaking discovery
will not bring down the price of groceries :(
 
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  • #2
Wilmer said:
Just discovered that:
if n is a 4 digit number such that the 2 digit number
formed by its 1st 2 digits is 1 greater than the
2 digit number formed by its last 2 digits,
and n is a square, then only one such n exists: 8281

Unfortunately, such a mathshaking discovery
will not bring down the price of groceries :(
Okay, so prove it's unique! And no making a big list of trial and error or you'll get coal in your stocking.

-Dan
 
  • #3
I found it...you prove it...why should I have all the fun!
 
  • #4
Wilmer said:
I found it...you prove it...why should I have all the fun!
I can prove it. It's intuitively obvious. (Quote from my Stat Mech professor.)

-Dan
 
  • #5
topsquark said:
I can prove it. It's intuitively obvious. (Quote from my Stat Mech professor.)

-Dan
Proof below
let the 2nd 2 digits be n so 1st 2 digits (n+1) and number be $x^2$

$100(n+1) + n = 101n + 100 = x^2$

now $101n = x^2 - 100 = (x+10)(x-10)$ and as 101 > n and 101 is prime so

x + 10 = 101k as as x is 2 digit number k =1 so x = 91 and we get the number $91^2=8281$
 
  • #6
No...no...
a = 1st digit, b = 2nd digit:

1010a + 101b - 1 = x^2 :)
 

FAQ: Amazing Math Discovery: 8281 is the Only 4 Digit Square with this Quirk

1. How was the discovery of 8281 being the only 4-digit square with this quirk made?

The discovery was made through a combination of mathematical analysis and computer programming. The researcher noticed a pattern in the squares of 4-digit numbers and used computer algorithms to test and confirm the uniqueness of 8281.

2. What makes the number 8281 so special compared to other 4-digit squares?

The number 8281 has a quirk that sets it apart from other 4-digit squares. When the digits of 8281 are multiplied in pairs (8*2 and 8*1), the resulting products are also consecutive numbers (16 and 8). This is the only 4-digit square with this property.

3. Is there a mathematical explanation for why 8281 is the only 4-digit square with this quirk?

Yes, there is a mathematical explanation for this phenomenon. It involves the properties of perfect squares and the relationship between digits in a number and their placement within the number.

4. Are there any practical applications for this discovery?

While this discovery may not have direct practical applications, it contributes to our understanding of the properties of numbers and can inspire further mathematical research and exploration.

5. Has this discovery been verified by other scientists?

Yes, this discovery has been verified by multiple scientists and mathematicians using various methods and algorithms. It has also been published in a peer-reviewed mathematical journal, adding to its credibility.

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