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Number Theory and Algebra Question

will someone please help me with this question for university?

Prove that 10^(3n)+1 cannot be represented as a sum of the cubes of two integers

Appreciate thanks. Inbox would be helpful ! :smile:
Hello, I've moved this to maths study help for you. :smile:
Reply 2
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Zacken
22
Original post by Lauren_ellams
will someone please help me with this question for university?

Prove that 10^(3n)+1 cannot be represented as a sum of the cubes of two integers

Appreciate thanks. Inbox would be helpful ! :smile:


Take n=1,b=1,a=10n=1, b=1, a=10 so: 103+1=a3+b310^3 + 1 = a^3 + b^3. Did you mean 103n+110^{3n+1}?
Original post by Lauren_ellams
will someone please help me with this question for university?

Prove that 10^(3n)+1 cannot be represented as a sum of the cubes of two integers

Appreciate thanks. Inbox would be helpful ! :smile:


10^(3n) + 1 = (10^n)^3 + 1^3, so it can be represented as the sum of two cubes.
The question is wrong.
Original post by Zacken
Take n=1,b=1,a=10n=1, b=1, a=10 so: 103+1=a3+b310^3 + 1 = a^3 + b^3. Did you mean 103n+110^{3n+1}?


yeah I did mean 10^(3n+1) my teacher gave the wrong question. could you give me a full explanation as it is worth 15 marks and I have no idea what to do!
Reply 5
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Zacken
22
Original post by Lauren_ellams
yeah I did mean 10^(3n+1) my teacher gave the wrong question. could you give me a full explanation as it is worth 15 marks and I have no idea what to do!


Work (mod7)\pmod{7}:

103n+1≡33n+1(mod7)≡3×27n(mod7)≡(−1)n×3(mod7)≡±3(mod7)10^{3n+1} \equiv 3^{3n+1} \pmod{7} \equiv 3 \times 27^n \pmod{7} \equiv (-1)^n \times 3 \pmod{7} \equiv \pm 3 \pmod{7}

What are cubes (mod7)\pmod{7}, hence what are sums of two cubes (mod7)\pmod{7} are they ever ±3\pm 3?
i don't know it doesn't say that is the question:/
Reply 7
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Zacken
22
Original post by Lauren_ellams
i don't know it doesn't say that is the question:/


What?
Original post by Lauren_ellams
i don't know it doesn't say that is the question:/


First of all, do you understand modular arithmetic?
Original post by HapaxOromenon
First of all, do you understand modular arithmetic?

yeah but it is so confusing. it has to be proven algebraically
Original post by Lauren_ellams
yeah but it is so confusing. it has to be proven algebraically


Read this: https://nrich.maths.org/4350 to understand modular arithmetic before going any further.
Original post by Zacken
What?


I don't understand the question, can you please show me the answer??? :frown:
Reply 12
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Zacken
22
Original post by Lauren_ellams
I don't understand the question, can you please show me the answer??? :frown:


It's a proof... there is no "answer".
the proof then
Reply 14
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Zacken
22
Original post by Lauren_ellams
the proof then


I've given you the outline, have you bothered reading it?
yeah but I don't understand
Reply 16
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Zacken
22
Original post by Lauren_ellams
yeah but I don't understand


What part don't you understand?
Original post by Zacken
What part don't you understand?


all of it
Original post by Zacken
What part don't you understand?


Original post by Lauren_ellams
all of it


this has been a very amusing thread

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