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Goldbach's conjecture states that every even number greater than two can be expressed as the sum of two prime numbers. For example, $2022 = 191 + 1831.$ How many ordered pairs of prime numbers have a sum of $100 \,?$

 Jul 11, 2023
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Goldbach's conjecture states that every even number greater than two can be expressed as the sum of two prime numbers. For example, $2022 = 191 + 1831.$ How many ordered pairs of prime numbers have a sum of $100 \,?$   

 

You could list all the prime numbers up to 50. 

No need to go past 50.   I counted 6 of them.  

 

  3,  11,  17,  29,  41,  47  

97,  89,  83,  71,  59,  53

 

 

Can't count 2 because 98 isn't prime.  

ditto   5 ... 95  

ditto   7 ... 93

ditto 13 ... 87  

ditto 19 ... 81  

ditto 23 ... 77  

ditto 31 ... 69  

ditto 37 ... 63

ditto 43 ... 57  

 

edited to add ~  I'm not certain what an "ordered" pair is. 

                          So if the sequence matters, then it's 12.  

.

 Jul 11, 2023
edited by Bosco  Jul 11, 2023
 #2
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WLOG 

 

realize for two numbers to sum up to 100, one must be < 50 (as 50 isn't prime)

 

There are 6 prime numbers less than 50 that satisfy our condition

And these 6 prime numbers could be placed in the x coord or the y coord

 

So 6*2= 12 

 Jul 12, 2023

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