MHB Solve Equation II: $(1+a!)(1+b!)=(a+b)!$

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The equation $(1+a!)(1+b!)=(a+b)!$ is being discussed for solutions in non-negative integers. Participants are encouraged to contribute their findings and methods for solving it. The conversation highlights the importance of collaboration in tackling complex mathematical problems. Engagement from users is noted, with appreciation for contributions. The thread emphasizes the collective effort in finding solutions to the equation.
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Solve in non-negative integers the equation $(1+a!)(1+b!)=(a+b)!$.
 
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anemone said:
Solve in non-negative integers the equation $(1+a!)(1+b!)=(a+b)!$.

Hello.

(1+a!)(1+b!)=1+a!+a!b!+b!=(a+b)!

If \ a>b \rightarrow{}b!|(a!,a!b!,(a+b)!)

Therefore:

b!|1 \rightarrow{}b!=1 \rightarrow{}b=0 \ or \ b=1

If \ b=1 \rightarrow{}(1+a!)2=(a+1)!

2+2a!=(a+1)a! \rightarrow{}a=2

If \ b=0 \rightarrow{}(1+a!)2=a!

a!=-2 \ absurdity

The end:

If \ a=b \rightarrow{}(1+a!)(1+a!)=1+2a!+a!a!=(2a)!

a!|1 \rightarrow{}a=0 \ or \ a=1

If \ a=0 \rightarrow{}2*2=1 \ absurdity

If \ a=1 \rightarrow{}2*2=2 \ absurdity

I will conclude by:

If \ a>b \rightarrow{}a=2 \ and \ b=1

Same b>a ...

Regards.
 
Last edited:
mente oscura said:
Hello.

(1+a!)(1+b!)=1+a!+a!b!+b!=(a+b)!

If \ a>b \rightarrow{}b!|(a!,a!b!,(a+b)!)

Therefore:

b!|1 \rightarrow{}b!=1 \rightarrow{}b=0 \ or \ b=1

If \ b=1 \rightarrow{}(1+a!)2=(a+1)!

2+2a!=(a+1)a! \rightarrow{}a=2

If \ b=0 \rightarrow{}(1+a!)2=a!

a!=-2 \ absurdity

The end:

If \ a=b \rightarrow{}(1+a!)(1+a!)=1+2a!+a!a!=(2a)!

a!|1 \rightarrow{}a=0 \ or \ a=1

If \ a=0 \rightarrow{}2*2=1 \ absurdity

If \ a=1 \rightarrow{}2*2=2 \ absurdity

I will conclude by:

If \ a>b \rightarrow{}a=2 \ and \ b=1

Same b>a ...

Regards.

Well done, mente oscura! Thanks for participating too!
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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