Fermat Number Puns

Hey everyone, I've been stuck on a homework question for about 2 days, and I'd wonder if any of you have any idea how to prove this : Suppose x,y,z are coprime and are solution of x^3 +y^3 =z^3. Show xy(x+y) is divisible by 3 Hint : use fermat's little theorem I think the answer shouldn't be hard, yet I just can't find it. It is a subquestion to show FLT for n=3 Thank you for your help

Source Image.

The question firmly states

"Let F_n = 2^2^n +1 be the nth Fermat number. Prove that F_n|2^Fn - 2 for all n >= 1".

I understand the proof uptil it says "It suffices therefore to show that 2^n+1 | Fn - 1, or equivalently n + 1 <= 2^n". I am comfortable in reasoning why F_n | (2^2^n +1 ) (2^2^n -1) . Just the following line is unclear

But I am having a hard time exactly following this line . I don't get ""It suffices therefore to show that 2^n+1 | Fn - 1, or equivalently n + 1 <= 2^n"." .
I tried solving it myself (using contradiction) but failed so if there is a alternate proof please help !

P.S: This is not my homework . I am learning Number Theory on my own . I am fairly new to it

I know that if p is prime and a is not divisible by p then this congruency holds.

But I'm not able to understand how and why exactly the algorithm works. Can someone please explain it in a simple way? How would I check whether a given number n, where 1 <= n <= 10^(12) is prime?

(To the tune of “a doe a deer”)

If p, a prime,

Does not divide a,

Which is a positive integer,

Then p divides,

a to the p,

Minus 1, minus 1.

Is the converse also true?

No, not necessarily.

Does that thought make you feel blue?

No, ‘cause pseudoprimes are cool!

According to Simon Singh's book "Fermat Last Theorem" ( https://www.amazon.co.uk/Fermats-Last-Theorem-Confounded-Greatest/dp/1841157910 - highly recommended by the way), Fermat proved that 26 is the only number sandwiched between a square and a cube.

How would you go about proving this?

What tools did Fermat have available to him in order to solve this?

I am just interested in a general discussion of how people approach this.

My personal approach is working in mod(4) and mod(3) and try to deduce a few things - but i haven't been able to spend much time on it yet.

So Fermat had a theorem about the sum of two squares equaling a prime number if said prime number has a remainder of 1 when divided by 4. And it seems to be true for numbers other than prime numbers.

Well, it made me think a bit about something.

Since it is known whether or not a number can be expressed as the sum of two squares just by dividing by 4, is there something like that, for say, the sum of two cubes? What about the sum of three cubes? Or, more generally, something like this:

Is there a way to check whether a number can be expressed as k numbers each raised to the nth power? And what about k = n?

Hi there,

I am totally lost on how to prove the following statement:

If n is not a power of 2, then 2^n + 1 is not a prime. I tried playing around with factoring something of the form x^m+y^m but realized quickly that's easier said than done.

The problem is to prove 24^31 is congruent to 23^32 (mod 19).

Is the point of FLT that you can manipulate large numbers and prove their congruences?

I honestly don't know how to proceed with this problem. At all. I found people who also worked on this problem on stackoverflow (http://math.stackexchange.com/questions/1768700/prove-that-2431-is-congruent-to-2332-mod-19) but I don't really understand what's happening step by step. Could some break this down for me?

{199 763 108 369 144 246 764 874 350 589 060 318 327 913 757  214 481 373 615 302 041 \
  772 460 035 846 722 861 237 024 535,
 516 389 133 530 399 295 580 907 119 133 520 982 845 402 241 989 579 150 087 047 557 \
  500 252 382 877 946 879 694 574 908,
 503 916 542 857 420 549 161 320 504 331 569 885 163 106 303 311 791 247 568 290 158 \
  150 592 400 078 487 478 623 955 569,
 703 634 292 019 309 468 474 431 206 533 076 036 090 371 492 870 673 163 827 470 919 \
  214 030 250 637 835 149 094 669 020}

This message was explicitly said to be "an encrypted message" right under it. Above it was an image of a stamp featuring Fermat's last theorem (saying "x^n + y^n = z^n has no solution for integers n>2") but it was in a PDF so not easy to extract, therefore I doubt it has to be considered here. I tried running this in Python, checking if they are primes (they aren't), or if any permutation of them verifies x^n + y^n = z^n, none does, or if they individually form words in decimal ASCII/Unicode, or if their sum/difference does. It doesn't. By the way, if you end up on a sentence, it might be french. Try looking for little words like "le", "la", "un", "une", "de", "ou" etc. these are among the most commonly found. Also, the difficulty of this problem should not require much more than high school proficiency in math. It might be a little bit above but not by much.

Any idea ?

V sbyybjrq gur ehyrf

After explaining Fermat's little theorem, Recreations in the Theory of Numbers says: "In Mersenne's numbers, that is, numbers of the form 2^n - 1, the exponent n is always a prime and therefore odd (except for n = 2, a trivial case). Hence, since in Fermat's theorem p is also a prime and odd, it follows that mn must be even, and therefore m must be even, equal to, say 2r, and then p = 2rn+1. [Here is where I get confused] This tells us, therefore, that if, for example, 2^11 -1 has a prime divisor, p (if it has any divisor, it has a prime divisor, of course), it must be of thr form 2r·11+1 = 22r+1, and in fact when r = 1, the prime 23 divides 2^11 - 1."

I feel like I understand Fermat's little theorem, but I don't see how the conclusion follows from it. Why does Fermat's little theorem imply all prime divisors of 2^n - 1 (when n is prime) must be of the form 2rn+1?

I know the proof has been worked on for centuries, but now that Wiles' proof has been verified and awarded, I'm curious as to what impact this has for mathematics or science as a whole, besides satisfaction and some closure. Basically, what new fields of mathematics or ideas are available to us as a result?

Hi, this is my first question in this sub. We have a test tomorrow, and I just can't figure this out. (Sorry if formatting sucks, I'll try my best)

Problem: Find the last 2 digits of 39^39^390

My attempt:

39^39^390 ≡x (mod 100)

I know I'm supposed to use Fermat-Euler for 39, but I need clarification as to why?

So: 39^phi(100) ≡1 (mod 100)

Phi 100 is 40, so 39^40 ≡1 (mod 100)

But I have no idea where to go from here or what I even need this for.

Thanks for the help in advance.

FLINT headers:

#include "flint/flint.h" #include "flint/ulong_extras.h"

function:

static bool FermatProbablePrimalityTestFast(const mpz_class& n, unsigned int& nLength, CPrimalityTestParams& testParams, bool fFastFail = false) {

  mp_limb_t d = mpz_get_ui(n.get_mpz_t());

  if (n_is_probabprime_BPSW(d))
    return true;


  //if (mpz_probab_prime_p (n.get_mpz_t(), 12))
  //   return true;


// Faster GMP version
mpz_t& mpzE = testParams.mpzE;
mpz_t& mpzR = testParams.mpzR;

mpz_sub_ui(mpzE, n.get_mpz_t(), 1);
mpz_powm(mpzR, mpzTwo.get_mpz_t(), mpzE, n.get_mpz_t());
if (mpz_cmp_ui(mpzR, 1))
    return true;
if (fFastFail)
    return false;
// Failed Fermat test, calculate fractional length
mpz_sub(mpzE, n.get_mpz_t(), mpzR);
mpz_mul_2exp(mpzR, mpzE, nFractionalBits);
mpz_tdiv_q(mpzE, mpzR, n.get_mpz_t());
unsigned int nFractionalLength = mpz_get_ui(mpzE);
if (nFractionalLength >= (1 << nFractionalBits))
    return error("FermatProbablePrimalityTest() : fractional assert");
nLength = (nLength & TARGET_LENGTH_MASK) | nFractionalLength;
return false;

}

and

static bool FermatProbablePrimalityTest(const CBigNum& n, unsigned int& nLength) { CAutoBN_CTX pctx; CBigNum a = 2; // base; Fermat witness CBigNum e = n - 1; CBigNum r;

if (n_is_probabprime_BPSW(n.getulong()))
    return true;

BN_mod_exp(&r, &a, &e, &n, pctx);
if (r == 1)
    return true;

// Failed Fermat test, calculate fractional length
unsigned int nFractionalLength = (((n-r) << nFractionalBits) / n).getuint();
if (nFractionalLength >= (1 << nFractionalBits))
    return error("FermatProbablePrimalityTest() : fractional assert");
nLength = (nLength & TARGET_LENGTH_MASK) | nFractionalLength;
return false;

}

Can't see how to do this, any help would be appreciated

What is the mathematical correlation between the prime sum of two squares and imaginary numbers? Is there a math reason that they both exhibit (mod 4) characteristics?

~curious philosophy geek

x^n + y^n = z^n does not exist for n not 1. If not than

While x^n + y^n = z^n is unresolved hide n : Let Fermat Number n = x^n + y^n / z^n : if this is not resolved than let n be not 1 : call this a point.

Resolve this^I^t^^^is^obvious^this^is^a^continuous^task^so^call^n^and^be^not^1^.^AutocopyLove.

Anything raised to the same point is resolvable.

What's the point in continuing when you are hide?

But this is absurd to go on with while you can see this resolves itself continually.

Therefore: x^n + y^n = z^n has no whole number solution when n is greater than 2.

Trying to reduce this to get the result but I'm having a bit of trouble 'seeing' it.

Using Pepin's test 3^{(F_5-1)/2} = 3^(2^31) = 3^(2,146,483,648) = 10,324,303 ≢ -1 (mod 4,294,967,297)

Where F_5 is Fermat's 5th number

I asked this question yesterday in /r/math, but haven't gotten any responses so far, so I thought I'd come here! If we consider the set of numbers which are [; K_1 ;] -gonal and [; K_2 ;] -gonal, does there exist a simple expression (in terms of [; K_1 ;] and [; K_2 ;]) for the number of these values it takes summed to express any integer? If not, is there a subset of the integers for which such a relation can be defined (perhaps for n > some value m)?

Now consider the case where our addends must all belong to not only 2, but a set of d distinct polygonal number sets where d≥2. Can we still define such a subset (if necessary), of the integers such that these numbers can be rewritten as a sum of at most some function [; k( K_1 , K_2 ,..., K_d ) ;] terms? I realize this is vague, but I'm fairly uncertain about what the results of this kind of generalization would be, so this form is necessary. It is also possible that this would not be a generalization, but that such a statement could be made as a trivial consequence of Fermat's polygonal number theorem and the properties of polygonal numbers. What led me to wonder about such a generalization is the existence of many possible representations in terms of k k-gonal numbers for large n. It seems intuitive that there should be some stronger theorem without this runaway...

Another curiosity is the question of what similar properties are possessed by platonic numbers. Are there any theorems regarding platonic numbers that could be seen as analogous to Fermat's polygonal number theorem for the polygonal numbers? Of course, there being only 5 regular plantonic solids as opposed to infinite regular polygons, the "generalizations" I spoke of before applied to this problem could have only d≤5.

Any help you can give will be much appreciated!

Edit: Regarding the last part about a similar idea in 3 dimensions, Pollock made several conjectures for tetrahedral, octahedral, and cubic numbers, stating that any integer can be re-written as a sum of ≤ 5, 7, or 9 terms respectively. I believe the first two cases remain open problems.

If we consider the set of numbers which are [; K_0 ;] -gonal and [; K_1 ;] -gonal, does there exist a simple expression (in terms of [; K_0 ;] and [; K_1 ;]) for the number of these values it takes summed to express any integer? If not, is there a subset of the integers for which such a relation can be defined (perhaps for n > some value m)? Now consider the case where our addends must all belong to not only 2, but a set of d distinct polygonal number sets where d≥2. Can we still define such a subset (if necessary), of the integers such that these numbers can be rewritten as a sum of at most some function [; k( K_1 , K_2 ,..., K_d ) ;] terms? I realize this is vague, but I'm fairly uncertain about what the results of this kind of generalization would be, so this form is necessary.

Another curiosity is the question of what similar properties are possessed by platonic numbers. Are there any theorems regarding platonic numbers that could be seen an analogous to Fermat's polygonal number theorem for the polygonal numbers?

Edit: typ-ew Edit2: tried out some LaTeX