Calling all positive integers ...

[bonanova]

OK maybe not all.

How many pairs of mutually prime positive integers sum to 2x10²⁵⁰?

You should be able to do this on the back of an envelope, and probably can't count the pairs with a simple program (before the earth freezes.)

Pairs differing only in the order of addition count as 1 pair, not as 2 different pairs.

[witzar]

4x10

²⁴⁹

[superprismatic]

4x10²⁴⁹-1, here's why:

The pairs must contain only odd numbers, so this brings us down to 10²⁵⁰ possible
pairs. Furthermore, each pair must contain only those odd numbers which are not
equal to 0 modulo 5. This brings us to (4/5)x10²⁵⁰ which equals 8x10²⁴⁹. But,
this is 2 too many because the sequence of odd numbers begins (modulo 5) with
1,3,2,4,1,3,2,4... but ends in 1,3 which is the middle of this 4-long cycle.
Which brings us to 8x²⁴⁹-2 pairs. But this is double what we want because
it counts each pair twice, counting the order of addition both ways. Therefore,
we need to divide this in two to get 4x10²⁴⁹-1 pairs.

[dark_magician_92]

Agree with Superprismatic. I think u have counted (1, 2*10^250-1) also.

[bonanova]

4x10

²⁴⁹-1, here's why:

The pairs must contain only odd numbers, so this brings us down to 10²⁵⁰ possible

pairs. Furthermore, each pair must contain only those odd numbers which are not

equal to 0 modulo 5. This brings us to (4/5)x10²⁵⁰ which equals 8x10²⁴⁹. But,

this is 2 too many because the sequence of odd numbers begins (modulo 5) with

1,3,2,4,1,3,2,4... but ends in 1,3 which is the middle of this 4-long cycle.

Which brings us to 8x²⁴⁹-2 pairs. But this is double what we want because

it counts each pair twice, counting the order of addition both ways. Therefore,

we need to divide this in two to get 4x10²⁴⁹-1 pairs.

Consider the case of 2 x 10¹. Answer: 4 x 10⁰.

1 19

2 18

3 17

4 16

5 15

6 14

7 13

8 12

9 11

10 10

Suppose I were to argue that 2x10²⁵⁰ constructs an integral number of similar groups of 10 sums, each of which gives back 6 rows to the mutually prime condition.

Which group contributes the (-1)?

[superprismatic]

4x10

²⁴⁹-1, here's why:

The pairs must contain only odd numbers, so this brings us down to 10²⁵⁰ possible

pairs. Furthermore, each pair must contain only those odd numbers which are not

equal to 0 modulo 5. This brings us to (4/5)x10²⁵⁰ which equals 8x10²⁴⁹. But,

this is 2 too many because the sequence of odd numbers begins (modulo 5) with

1,3,2,4,1,3,2,4... but ends in 1,3 which is the middle of this 4-long cycle.

Which brings us to 8x²⁴⁹-2 pairs. But this is double what we want because

it counts each pair twice, counting the order of addition both ways. Therefore,

we need to divide this in two to get 4x10²⁴⁹-1 pairs.

Consider the case of 2 x 10¹. Answer: 4 x 10⁰.

1 19

2 18

3 17

4 16

5 15

6 14

7 13

8 12

9 11

10 10

Suppose I were to argue that 2x10²⁵⁰ constructs an integral number of similar groups of 10 sums, each of which gives back 6 rows to the mutually prime condition.

Which group contributes the (-1)?

It seems rather difficult to pull the wool over the Denizens!