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+--- Thread: Modular arithmetic (/showthread.php?tid=435)
Modular arithmetic - Stereotomy - 04/02/2010
I'll just preface this by saying I'm just a physics undergrad, so this might be a bit beyond my understanding, and I may well be missing something obvious or making a stupid mistake, but while playing around I noticed that it seems to be true that
\( ^{n}a\text{ mod }b= {}^{m}a\text{ mod }b \)
\( \text{For }^{n}a,{}^{m}a > b,\text{ }a,b,n,m \in \mathbb{N} \)
Is this actually true? And if so is there a proof of it I'll be able to wrap my mind around?
RE: Modular arithmetic - bo198214 - 04/02/2010
(04/02/2010, 07:16 PM)Stereotomy Wrote: \( ^{n}a\text{ mod }b= {}^{m}a\text{ mod }b \)
\( \text{For }^{n}a,{}^{m}a > b,\text{ }a,b,n,m \in \mathbb{N} \)
Is this actually true? And if so is there a proof of it I'll be able to wrap my mind around?
I dont think it is true. For example:
\( 7^7 \text{mod} 5 = 3\neq 2 = 7 \text{mod} 5 \)
RE: Modular arithmetic - Stereotomy - 04/03/2010
(04/02/2010, 10:57 PM)bo198214 Wrote:(04/02/2010, 07:16 PM)Stereotomy Wrote: \( ^{n}a\text{ mod }b= {}^{m}a\text{ mod }b \)
\( \text{For }^{n}a,{}^{m}a > b,\text{ }a,b,n,m \in \mathbb{N} \)
Is this actually true? And if so is there a proof of it I'll be able to wrap my mind around?
I dont think it is true. For example:
\( 7^7 \text{mod} 5 = 3\neq 2 = 7 \text{mod} 5 \)
Ah, good point, though
\( 7^{7^{7}} \text{mod} 5 = 3 \)
Is true as well. In fact, thinking about it, the numbers I tried out with this all had b>a. Perhaps that's an additional condition that either b > a or m, n > 1?
Just quickly tried this for a few low examples, a = 8, 9, 10, 11, and it seems to hold.
RE: Modular arithmetic - bo198214 - 04/03/2010
(04/03/2010, 12:18 AM)Stereotomy Wrote: \( 7^{7^{7}} \text{mod} 5 = 3 \)
Is true as well. In fact, thinking about it, the numbers I tried out with this all had b>a. Perhaps that's an additional condition that either b > a or m, n > 1?
There is an article which proves that \( {^n a} \) (which is \( a{\uparrow}^2 n \) in Knuth's arrow notation) finally will be constant for \( n\to\infty \) mod any \( M \), see this thread.