Sums of a positive cube and a power of 2

[enzocreti]

344 is an example of number that can be written as the sum of two positive cubes but also as the sum of a positive cube and a power of 2.
344=7^3+1=6^3+2^7

Are there other numbers with this property?

[mathwiz]

Originally Posted by enzocreti

344 is an example of number that can be written as the sum of two positive cubes but also as the sum of a positive cube and a power of 2.
344=7^3+1=6^3+2^7

Are there other numbers with this property?

Sure, for example:

344 = 1^3 + 7^3 = 6^3 + 2^7
1092728 = 1^3 + 103^3 = 94^3 + 2^18
2752 = 2^3 + 14^3 = 12^3 + 2^10
8741824 = 2^3 + 206^3 = 188^3 + 2^21
22016 = 4^3 + 28^3 = 24^3 + 2^13
69934592 = 4^3 + 412^3 = 376^3 + 2^24
176128 = 8^3 + 56^3 = 48^3 + 2^16
559476736 = 8^3 + 824^3 = 752^3 + 2^27
1729 = 9^3 + 10^3 = 12^3 + 2^0
4104 = 9^3 + 15^3 = 2^3 + 2^12
40033 = 9^3 + 34^3 = 33^3 + 2^12
24897817 = 9^3 + 292^3 = 201^3 + 2^24
39312 = 15^3 + 33^3 = 34^3 + 2^3
1409024 = 16^3 + 112^3 = 96^3 + 2^19
13832 = 18^3 + 20^3 = 24^3 + 2^3
32832 = 18^3 + 30^3 = 4^3 + 2^15
320264 = 18^3 + 68^3 = 66^3 + 2^15
199182536 = 18^3 + 584^3 = 402^3 + 2^27
4673088 = 25^3 + 167^3 = 164^3 + 2^18
67264 = 29^3 + 35^3 = 12^3 + 2^16
149389 = 29^3 + 50^3 = 53^3 + 2^9
314496 = 30^3 + 66^3 = 68^3 + 2^6
11272192 = 32^3 + 224^3 = 192^3 + 2^22
110656 = 36^3 + 40^3 = 48^3 + 2^6
262656 = 36^3 + 60^3 = 8^3 + 2^18
2562112 = 36^3 + 136^3 = 132^3 + 2^18
704977 = 41^3 + 86^3 = 89^3 + 2^3
3511872 = 41^3 + 151^3 = 152^3 + 2^6
37384704 = 50^3 + 334^3 = 328^3 + 2^21
684019 = 51^3 + 82^3 = 75^3 + 2^18
538112 = 58^3 + 70^3 = 24^3 + 2^19
1195112 = 58^3 + 100^3 = 106^3 + 2^12
2515968 = 60^3 + 132^3 = 136^3 + 2^9
90177536 = 64^3 + 448^3 = 384^3 + 2^25
1331064 = 67^3 + 101^3 = 110^3 + 2^6
885248 = 72^3 + 80^3 = 96^3 + 2^9
2101248 = 72^3 + 120^3 = 16^3 + 2^21
20496896 = 72^3 + 272^3 = 264^3 + 2^21
3375001 = 73^3 + 144^3 = 150^3 + 2^0
5639816 = 82^3 + 172^3 = 178^3 + 2^6
28094976 = 82^3 + 302^3 = 304^3 + 2^9
3375008 = 83^3 + 141^3 = 150^3 + 2^3
2048391 = 95^3 + 106^3 = 127^3 + 2^3
299077632 = 100^3 + 668^3 = 656^3 + 2^24
5472152 = 102^3 + 164^3 = 150^3 + 2^21
4304896 = 116^3 + 140^3 = 48^3 + 2^22
9560896 = 116^3 + 200^3 = 212^3 + 2^15
20127744 = 120^3 + 264^3 = 272^3 + 2^12
89576767 = 127^3 + 444^3 = 447^3 + 2^18

among many (presumably infinitely) more.

Python or PARI/GP is your friend...

[enzocreti]

Can you please give me your pari code?

[enzocreti]

Originally Posted by mathwiz

Sure, for example:

344 = 1^3 + 7^3 = 6^3 + 2^7
1092728 = 1^3 + 103^3 = 94^3 + 2^18
2752 = 2^3 + 14^3 = 12^3 + 2^10
8741824 = 2^3 + 206^3 = 188^3 + 2^21
22016 = 4^3 + 28^3 = 24^3 + 2^13
69934592 = 4^3 + 412^3 = 376^3 + 2^24
176128 = 8^3 + 56^3 = 48^3 + 2^16
559476736 = 8^3 + 824^3 = 752^3 + 2^27
1729 = 9^3 + 10^3 = 12^3 + 2^0
4104 = 9^3 + 15^3 = 2^3 + 2^12
40033 = 9^3 + 34^3 = 33^3 + 2^12
24897817 = 9^3 + 292^3 = 201^3 + 2^24
39312 = 15^3 + 33^3 = 34^3 + 2^3
1409024 = 16^3 + 112^3 = 96^3 + 2^19
13832 = 18^3 + 20^3 = 24^3 + 2^3
32832 = 18^3 + 30^3 = 4^3 + 2^15
320264 = 18^3 + 68^3 = 66^3 + 2^15
199182536 = 18^3 + 584^3 = 402^3 + 2^27
4673088 = 25^3 + 167^3 = 164^3 + 2^18
67264 = 29^3 + 35^3 = 12^3 + 2^16
149389 = 29^3 + 50^3 = 53^3 + 2^9
314496 = 30^3 + 66^3 = 68^3 + 2^6
11272192 = 32^3 + 224^3 = 192^3 + 2^22
110656 = 36^3 + 40^3 = 48^3 + 2^6
262656 = 36^3 + 60^3 = 8^3 + 2^18
2562112 = 36^3 + 136^3 = 132^3 + 2^18
704977 = 41^3 + 86^3 = 89^3 + 2^3
3511872 = 41^3 + 151^3 = 152^3 + 2^6
37384704 = 50^3 + 334^3 = 328^3 + 2^21
684019 = 51^3 + 82^3 = 75^3 + 2^18
538112 = 58^3 + 70^3 = 24^3 + 2^19
1195112 = 58^3 + 100^3 = 106^3 + 2^12
2515968 = 60^3 + 132^3 = 136^3 + 2^9
90177536 = 64^3 + 448^3 = 384^3 + 2^25
1331064 = 67^3 + 101^3 = 110^3 + 2^6
885248 = 72^3 + 80^3 = 96^3 + 2^9
2101248 = 72^3 + 120^3 = 16^3 + 2^21
20496896 = 72^3 + 272^3 = 264^3 + 2^21
3375001 = 73^3 + 144^3 = 150^3 + 2^0
5639816 = 82^3 + 172^3 = 178^3 + 2^6
28094976 = 82^3 + 302^3 = 304^3 + 2^9
3375008 = 83^3 + 141^3 = 150^3 + 2^3
2048391 = 95^3 + 106^3 = 127^3 + 2^3
299077632 = 100^3 + 668^3 = 656^3 + 2^24
5472152 = 102^3 + 164^3 = 150^3 + 2^21
4304896 = 116^3 + 140^3 = 48^3 + 2^22
9560896 = 116^3 + 200^3 = 212^3 + 2^15
20127744 = 120^3 + 264^3 = 272^3 + 2^12
89576767 = 127^3 + 444^3 = 447^3 + 2^18

among many (presumably infinitely) more.

Python or PARI/GP is your friend...

In the output of your Pari code I note that when the power of 2 is a prime greater than 3 then one of the positive cubes is a multiple of 7.
So for example
16^3+112^3=2^19+96^3. 19 is prime and 112 is a multiple of 7. Is so in general?

[enzocreti]

I mean
When the number can be written as the sum of a positive cube and a power of 2 (with the exponent prime greater than 3) then the number can be written as the sum of two positive cubes whose one is a multiple of 7.

344 can be written as the sum of 2^7+6^3...7 is a prime greater than 3. So 344 can be written also as the sum of two positive cubes whose one 7^3 is multiple of 7

[Dr Sardonicus]

Originally Posted by enzocreti

344 is an example of number that can be written as the sum of two positive cubes but also as the sum of a positive cube and a power of 2.
344=7^3+1=6^3+2^7

Are there other numbers with this property?

n^3 + 8 = n^3 + 2^3, any n.

Oh -- you want the power of 2 to be greater than 8?

n^3 + 64 = n^3 + 4^3 = n^3 + 2^6, any n.

n^3 + 512 = n^3 + 8^3 = n^3 + 2^9, any n.

...

[CRGreathouse]

T=thueinit('z^3+1);
is(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0;
list(lim)=my(v=List(),K=1,t); lim\=1; while(K<lim, for(n=1, sqrtnint(lim-K,3), if(is(t=K+n^3), listput(v,t))); K*=2); Set(v);

If I calculate correctly there are 24,666 examples below 10^10, the largest being 9998629153.