Mixing
For which even integers $k$ has $varphi(n+1)-varphi(n)=k$ a solution?
$begingroup$
For which even integers $k$ does the equation $$varphi(n+1)-varphi(n)=k$$ have a solution ?
$varphi(n)$ denotes the totient function and $n$ is a positive integer.
For the following $|k|le 1 000$, there is no solution $n$ in the range $[3,10^7]$ :
-958 -926 -910 -898 -892 -846 -834 -814 -790 -730 -682 -610 -594 -582 -570 -550
-514 -490 -462 -442 -422 -370 -354 -326 -310 -226 -202 -114 10 86 126 134 182 22
6 242 266 274 278 286 298 326 370 378 386 446 450 466 470 530 538 574 578 610 62
6 634 638 666 678 706 734 738 758 770 786 790 806 822 826 830 842 866 874 898 91
4 926 932 938 970 986
Almost all those numbers are of the form $ 4k+2$
In fact, the $|k|le 1 000$ divisible by $4$ having no solution in the range $[3,10^6]$ are $-892$ (solution $10 814 714$) and $932$ (no solution in the range $[3,10^8]$)
Does a solution exist for $k=932$ ?
number-theory elementary-number-theory integers totient-function
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
$endgroup$
add a comment |
$begingroup$
For which even integers $k$ does the equation $$varphi(n+1)-varphi(n)=k$$ have a solution ?
$varphi(n)$ denotes the totient function and $n$ is a positive integer.
For the following $|k|le 1 000$, there is no solution $n$ in the range $[3,10^7]$ :
-958 -926 -910 -898 -892 -846 -834 -814 -790 -730 -682 -610 -594 -582 -570 -550
-514 -490 -462 -442 -422 -370 -354 -326 -310 -226 -202 -114 10 86 126 134 182 22
6 242 266 274 278 286 298 326 370 378 386 446 450 466 470 530 538 574 578 610 62
6 634 638 666 678 706 734 738 758 770 786 790 806 822 826 830 842 866 874 898 91
4 926 932 938 970 986
Almost all those numbers are of the form $ 4k+2$
In fact, the $|k|le 1 000$ divisible by $4$ having no solution in the range $[3,10^6]$ are $-892$ (solution $10 814 714$) and $932$ (no solution in the range $[3,10^8]$)
Does a solution exist for $k=932$ ?
number-theory elementary-number-theory integers totient-function
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
$endgroup$
$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33
add a comment |
$begingroup$
For which even integers $k$ does the equation $$varphi(n+1)-varphi(n)=k$$ have a solution ?
$varphi(n)$ denotes the totient function and $n$ is a positive integer.
For the following $|k|le 1 000$, there is no solution $n$ in the range $[3,10^7]$ :
-958 -926 -910 -898 -892 -846 -834 -814 -790 -730 -682 -610 -594 -582 -570 -550
-514 -490 -462 -442 -422 -370 -354 -326 -310 -226 -202 -114 10 86 126 134 182 22
6 242 266 274 278 286 298 326 370 378 386 446 450 466 470 530 538 574 578 610 62
6 634 638 666 678 706 734 738 758 770 786 790 806 822 826 830 842 866 874 898 91
4 926 932 938 970 986
Almost all those numbers are of the form $ 4k+2$
In fact, the $|k|le 1 000$ divisible by $4$ having no solution in the range $[3,10^6]$ are $-892$ (solution $10 814 714$) and $932$ (no solution in the range $[3,10^8]$)
Does a solution exist for $k=932$ ?
number-theory elementary-number-theory integers totient-function
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
$endgroup$
For which even integers $k$ does the equation $$varphi(n+1)-varphi(n)=k$$ have a solution ?
$varphi(n)$ denotes the totient function and $n$ is a positive integer.
For the following $|k|le 1 000$, there is no solution $n$ in the range $[3,10^7]$ :
-958 -926 -910 -898 -892 -846 -834 -814 -790 -730 -682 -610 -594 -582 -570 -550
-514 -490 -462 -442 -422 -370 -354 -326 -310 -226 -202 -114 10 86 126 134 182 22
6 242 266 274 278 286 298 326 370 378 386 446 450 466 470 530 538 574 578 610 62
6 634 638 666 678 706 734 738 758 770 786 790 806 822 826 830 842 866 874 898 91
4 926 932 938 970 986
Almost all those numbers are of the form $ 4k+2$
In fact, the $|k|le 1 000$ divisible by $4$ having no solution in the range $[3,10^6]$ are $-892$ (solution $10 814 714$) and $932$ (no solution in the range $[3,10^8]$)
Does a solution exist for $k=932$ ?
number-theory elementary-number-theory integers totient-function
number-theory elementary-number-theory integers totient-function
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
asked Jan 25 at 12:47
PeterPeter
48.8k1239136
48.8k1239136
$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33
add a comment |
$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33
$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33
$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33
add a comment |
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$begingroup$
It appears that the numbers $|$$-$$892| = 892$ and $|932| = 932$ have an interesting property common to them. $$892+4=varphi(1347)=varphi(1856).$$ Notice that $(1347+1)^3=15^3+921^3+1186^3$ and $(1856+1)^3= 50^3+1065^3+1732^3$. These are the only sum-of-three-cube solutions for them both, respectively. Now: $$932+4=varphi(1027)=varphi(1431).$$ Notice that $(1027+1)^3=136^3+191^3+1025^3$ and $(1431+1)^3=435^3+786^3+1333^3$. These are also the only sum-of-three-cube solutions for them both, respectively. Does this mean anything (to you at least)? I do not know.
$endgroup$
– user477343
Feb 2 at 9:33