Question

Prove that every odd integer is the difference of two squares

Answer 1

i suppose best proof to use would be contradiction...

Answer 2

so i assume

x^2 - y^2 = 2k

Answer 3

then... for x^2 - y^2 to be even... both must be even or odd

Answer 4

so if x^2 and y^2 are both even... x and y are also even

Answer 5

no we're trying to prove converse i think, which doesnt justify the actual statement

Answer 6

so i can rewrite this...

(2m)^2 - (2n)^2 = 2k

Answer 7

4m^2 - 4n^2 = 2k

Answer 8

so....

2m ^ 2 - 2n^2 = k

Answer 9

then...

m^2 - n^2 = k/2

Answer 10

can i assume this is contradiction because 1/2 is not an integer?

Answer 11

could u plz clarify, you want to prove difference of two squares is alwasy an odd number, or you want to prove every odd number is a difference of two squares ?

Answer 12

no idea

Answer 13

how do you understand the question?

Answer 14

the question is, every odd integer can be written as difference of squares

Answer 15

but i see you're trying to prove its converse, which is not correct

Answer 16

so how would you do the assumption?

Answer 17

using contradiction

Answer 18

personally, i see no difference in solution

Answer 19

consider below two statements :

1) all odd numbers are diff of squares

2) all diff of squares are odd numbers

Answer 20

both are converses of each other

Answer 21

proving by contradiction is different

Answer 22

so tell me...can you think of a different solution then?

Answer 23

what would you equate 2k + 1 to in proof by contradiction then?

Answer 24

if you think about it...every difference of two squares is also odd...

Answer 25

first, did u get that a converse is not same as the actual statement

proving a converse doesnot justify the actual statement

Answer 26

i know...but i cannot see any other way to prove by contradiction....so if you have an idea...i'll gladly hear it

Answer 27

i think , it means :
(2n-1) = n^2 - (n-1)^2
just to expand right side, it can be a prove

Answer 28

@RadEn contradiction

Answer 29

then dont try to prove the converse, lets think of how to prove by contradiction, by assuming the opposite of what we need to prove :)

Answer 30

what is the opposite of difference of two squares? sum??

Answer 31

we can factor difference of squares

Answer 32

then you're using direct proof

Answer 33

doesnt look obvious to me the contradiction proof

Answer 34

looks involved

Answer 35

involved?

Answer 36

yes atleast to meh il give a try

Answer 37

i think you'll arrive with the same proof i was doing earlier...

Answer 38

thats not a proof

Answer 39

it is actually...probably not contradiction...but it is

Answer 40

it is a proof*

Answer 41

thats not a proof for this question, it is a proof for the converse of this question

Answer 42

you're welcome to give it a try