Question

Advanced level problem

Answer 1

IF x.y,z are prime numbers such that x=y^z+1 then the largest prime number by which x+10 y+20 z is divisible by

Answer 2

@Miracrown @jim_thompson5910 @ganeshie8 @nincompoop

Answer 3

THE ANSWER IS 13 EVERYONE BUT I WANT THE SOLLUTION AND EXACTLY HOW TO APPROACH THE PROBLEM

Answer 4

It is \[\huge x=y ^{z} +1 \]

Answer 5

we want to find largest prime possible that divides the sum of
p1+5*2p2+5*2^2p3
where p1=p2^p3+1
what can we say about the largest prime factor for the sum of a linear combination of 3 prime numbers

Answer 6

lets take the 5 and the 2 itself that should divide these 3 terms

Answer 7

5^2+1=26, which has 13 as its largest rime factor

Answer 8

now thats the easy part

Answer 9

you have to prove that there are no other larger prime numbes that divide these 3 linear comibnation of primes

Answer 10

which im working on right now, its a slightly long proof, maybe im going about it wrong

Answer 11

I see , so you got 13 and you have to prove no other prime numbers exist to divide that stuff

Answer 12

i will work on this later okay ill tell you if i get anything good, we should let ganeshie or someone try it

Answer 13

@ganeshie8

Answer 14

yes

Answer 15

i know xD

Answer 16

it was my postuolate that the factor of that must factor the sum of the other 3

Answer 17

this is number theory?

Answer 18

yah

Answer 19

@ParthKohli

Answer 20

if u ask me why my only answer is im nuts

Answer 21

i heard a voice say to do this

Answer 22

x+10 y+20 z = y^z + 1 + 10y + 20z = y(y^(z-1)-10) + 20z + 1

Answer 23

@ganeshie8 can solve any math problem in the world

Answer 24

i know

Answer 25

ugh i have no idea how to solve this lol

Answer 26

close any reopen this question in a new page, so someone will do it

Answer 27

too much spam here now

Answer 28

Yeah this question got my mind **** off

Answer 29

OK @dan815

Answer 30

next below facts may be helpful :
\(a^{p-1} \equiv 1 \mod p\)

\(abc+1\) is not divisible by none of a, b, or c if a,b,c are prime numbers

Answer 31

WE don't have all this stuff , the fermat theorem and all

Answer 32

no that is old statemest used in part of the infinite prime proof

Answer 33

\oh

Answer 34

Since its an advanced problem, it surely deserves an advanced solution :P
lets try all that we got first may be :)

Answer 35

POSTING THIS AS A NEW QUESTION <--------

Answer 36

x = y^z +1 and x is a prime. The posibilities:
y^z is odd iff y^z =1 to have x =2 but 1 is not a prime --> reject
y^z is even iff y =z =2 to have x =5 is a prime,
for this option, as I posted and deleted (hihihi) we have 5+20+40 =65 = 13*5
so that the largest prime is 13.
@No.name hehehe...

Answer 37

can you elaborate a little

Answer 38

you see, x must be a prime, and x = something +1
what is something when unless 2, all prime is odd?

Answer 39

odd+1 = even ,
and only even prime (2) +1 =odd

Answer 40

Yess

Answer 41

is there anyone understand me?? hehehe. my bad English!!
so?? read my proof

Answer 42

Wait one min , i think you have got it dude i am reading it . you are great

Answer 43

I mean to have y^z even, and y, z are prime, you have only one posibility: y =2,

Answer 44

I got the odd part not the even one

Answer 45

but the question is
then the largest prime number by which x+10 y+20 z is divisible by

Answer 46

Yes, I am trying to put it in neat

Answer 47

oh he solved it by putting values
he found the values of x,y,z from x=y^z+1

Answer 48

now he is saying that x,y,z are prime
now to get prime number as an answer in y^z+1 he is considering prime numbers

Answer 49

though this one is more practical that theoretical

Answer 50

YES! but a very nice approach @Loser66 u r great.
I shall post the answer when i discuss this with my professor

Answer 51

and the sollution

Answer 52

and you know that the prime numbers greater than 2 in power give odd number as an answer ,so if you add 1 in it
it will give u even number

Answer 53

oddxodd = odd
+1 would be even
which isnt x

Answer 54

But i think problems like these should only have a trial and error method . not a perfect sollution based method

Answer 55

YES ^

Answer 56

well ask for theoretical explaination tho,not all questions can be solved practicaly

Answer 57

Okay...:)

Answer 58

thank you @aajugdar

Answer 59

wow ! thats brilliant :)

Answer 60

yes nice logic

Answer 61

odd+1 = even ,
and only even prime (2) +1 =odd

did all the magic xD

Answer 62

haha yes

Answer 63

But seeing a question like this for the first time , and such a sweet and cute sollution too

Answer 64

ikr... :) i like below quote on NT :

Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics" (Beiler 1966, Goldman 1997).

Answer 65

The same thing we have in our book printed on the front page

Answer 66

Are you using Burton's number theory book ?

Answer 67

We get a booklet on every topic in math , and there is a most famous quote made by mathematicians on that particular topic on every page

Answer 68

So , there are like sums ranging from basic level to advanced level for every topic

Answer 69

and we get 50 such booklets in 2 years , (50 quotes hehehe , they are nice btw)

Answer 70

Gotcha :) 50 problem sets !! ?? thats part of JEE preparation is it ?

Answer 71

Yes , each problem set contains 600 problems

Answer 72

thats just ridankulous - you wont have a life next 2 years lol

Answer 73

Yes i have to make a lot of sacrifices,
There are other subjects too , physics and chem

Answer 74

Getting into IIT has now become like itself a Sacrifice lol... You have to sacrifice your hobbies too! :(

Answer 75

I recently got a booklet for sequences and series so now i will be flooding OS with sequence questions

Answer 76

It also has a quote

Answer 77

@mathslover yes but imagine the life after getting into the IIt's it makes me work harder

Answer 78

NO gains without pains

Answer 79

you stay in kota right math

Answer 80

No @No.name

Answer 81

okay

Answer 82

i wish to share a link on number theory with you, ganeshie gave me do u wannt it

Answer 83

@mathslover

Answer 84

Yes sure. I will love to have a look at it.

Answer 85

Okay. Thanks a lot.

Answer 86

I it is not working @ganeshie8 the link u gave me

Answer 87

hey @No.name its a pirated version - thats the reason i msged u :)
u need to have torrent software to download it...

delete the link...

Answer 88

lol ^

Answer 89

here is the book if u want to buy :
http://www.amazon.com/Elementary-Number-Theory-David-Burton/dp/0073383147/ref=cm_lmf_tit_9/179-6881008-2171860

Answer 90

Deleted it

Answer 91

ty :D

Answer 92

lol the price

Answer 93

haha need to take a loan or steal a bank to buy it :P

Answer 94

he is still in 11th grade,so

Answer 95

Yeah so i would not got to jail

Answer 96

Even if i rob a bank