1 + 9 = 10 36 + 64 = 100 Is there a pair of squares for 1,000; 10,000; 100,000 etc. ? Is there an upper limit ? Are there any gaps ? What if we start with a different modulus, ie., 5 so 1 + 4 = 5 and 9 + 16 = 25. Remember, a modulus can be greater than 10. For example, with a modulus of 13, 13=10, 169=100; 2197=1000. Which moduli produce pair of squares adding to 10, 100 etc. ? What if higher powers are used, ie., exponent = 3, 4, 5, or ........ ? Why do these sequences of pairs exist ?
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Moduli?
Wait I almost read that as 'ie., 51+4=59+16=25'
so what it's supposed to be is 'ie., 5 {1+4=5} {9+16=25}' right?
:)
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Moduli ?
Hi Rebecca,
You're right & I've been sloppy. It's best if you write it in both base 10 and base 5.
5 {1 + 4 = 5} { 9 + 16 = 25}
{1 + 4 = 10} {14 + 31 = 100}
For mod 5, there are also squares that sum to 1,000 (125 in mod 10), 10,000 (625), 100,000 (3125), a million (15,625), 10 million (78,125) and 100 million (390,625). I have not investigated any further.
I look forward to seeing what you have found out. Read my suggestions on the original posting to see a quick way to get lots of data (if you have excel & know some simple programming in it).
Nicholas