## Pandigital numbers | ArithmeticMean[],GeometricMean[]

Each of the positive integers   $A, \; B, \; C, \; D$   is pandigital such that

>>   $A \; < \; B \; < \; C \; < \; D$
>>   $A \; + \; D$   is also pandigital
>>   $B \; = \; GeometricMean \,[A, \; D]$
>>   $C \; = \; ArithmeticMean \,[A, \; D]$

Find   $A, \; B, \; C, \; D$

Here's one solution:

$A \; = \; 1076539482$

$B \; = \; 2153078964 \; = \; GeometricMean \,[ \,1076539482, \; 4306157928 \,]$

$C \; = \; 2691348705 \; = \; ArithmeticMean \,[1076539482, \; 4306157928 \,]$

$D \; = \; 4306157928$

$A \; + \; D \; = \; 1076539482 \; + \; 4306157928 \; = \; 5382697410$

Each of the integers   $A, \; B, \; C, \; D$   is pandigital

Find other solutions.

Paul found:

where   $A + D$   is ten times a pandigital number:

$\{2107653948,4215307896,5269134870,8430615792,10538269740 \}$
$\{2136507948,4273015896,5341269870,8546031792,10682539740 \}$
$\{2136759048,4273518096,5341897620,8547036192,10683795240 \}$
$\{2147936508,4295873016,5369841270,8591746032,10739682540 \}$
$\{2150793648,4301587296,5376984120,8603174592,10753968240 \}$
$\{2197530864,4395061728,5493827160,8790123456,10987654320 \}$
$\{2409513678,4819027356,6023784195,9638054712,12047568390 \}$

And where   $A + D$   is a pandigital number:

$\{1076539482,2153078964,2691348705,4306157928,5382697410 \}$
$\{1098765432,2197530864,2746913580,4395061728,5493827160 \}$
$\{1234567890,2469135780,3086419725,4938271560,6172839450 \}$
$\{1234568790,2469137580,3086421975,4938275160,6172843950 \}$
$\{1269835704,2539671408,3174589260,5079342816,6349178520 \}$
$\{1270356984,2540713968,3175892460,5081427936,6351784920 \}$
$\{1274590368,2549180736,3186475920,5098361472,6372951840 \}$
$\{1298435670,2596871340,3246089175,5193742680,6492178350 \}$
$\{1365079482,2730158964,3412698705,5460317928,6825397410 \}$
$\{1367590482,2735180964,3418976205,5470361928,6837952410 \}$
$\{1472590368,2945180736,3681475920,5890361472,7362951840 \}$
$\{1476859032,2953718064,3692147580,5907436128,7384295160 \}$
$\{1479365082,2958730164,3698412705,5917460328,7396825410 \}$
$\{1564329870,3128659740,3910824675,6257319480,7821649350 \}$
$\{1587296430,3174592860,3968241075,6349185720,7936482150 \}$
$\{1274590368,3186475920,3186475920,5098361472,6372951840 \}$
$\{1507936482,3015872964,3769841205,6031745928,7539682410 \}$
$\{1526437098,3052874196,3816092745,6105748392,7632185490 \}$
$\{1642958730,3285917460,4107396825,6571834920,8214793650 \}$
$\{1705483926,3410967852,4263709815,6821935704,8527419630 \}$
$\{1728439506,3456879012,4321098765,6913758024,8642197530 \}$
$\{1759048632,3518097264,4397621580,7036194528,8795243160 \}$
$\{1472590368,3681475920,3681475920,5890361472,7362951840 \}$
$\{1847356290,3694712580,4618390725,7389425160,9236781450 \}$
$\{1950637284,3901274568,4876593210,7802549136,9753186420 \}$
$\{1962530874,3925061748,4906327185,7850123496,9812654370 \}$
$\{1962730854,3925461708,4906827135,7850923416,9813654270 \}$
$\{1962850734,3925701468,4907126835,7851402936,9814253670 \}$
$\{1975308642,3950617284,4938271605,7901234568,9876543210 \}$

$A \; + \; D \; = \; B$
$ArithmeticMean \, [A, \; D] \; = \; C$
$GeometricMean \, [A, \; D] \; = \; E$

Format   $\{A, \; B, \; C, \; D, \; E\}$

$\{1098765432,5493827160,2746913580,4395061728,2197530864\}$
$\{1269835704,6349178520,3174589260,5079342816,2539671408\}$
$\{1270356984,6351784920,3175892460,5081427936,2540713968\}$
$\{1274590368,6372951840,3186475920,5098361472,2549180736\}$
$\{1472590368,7362951840,3681475920,5890361472,2945180736\}$
$\{1476859032,7384295160,3692147580,5907436128,2953718064\}$
$\{1098765432,7691358024,2746913580,4395061728,2197530864\}$
$\{1098765432,8790123456,2746913580,4395061728,2197530864\}$
$\{1759048632,8795243160,4397621580,7036194528,3518097264\}$
$\{1950637284,9753186420,4876593210,7802549136,3901274568\}$
$\{1234567890,9876543120,3086419725,4938271560,2469135780\}$

math grad - Interest: Number theory
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### 5 Responses to Pandigital numbers | ArithmeticMean[],GeometricMean[]

1. paul says:

Here are all of them ( can’t guarantee that though 🙂 ).

Format is {a, b, c, d}.

{1076539482,2153078964,2691348705,4306157928}
{1098765432,2197530864,2746913580,4395061728}
{1234567890,2469135780,3086419725,4938271560}
{1234568790,2469137580,3086421975,4938275160}
{1269835704,2539671408,3174589260,5079342816}
{1270356984,2540713968,3175892460,5081427936}
{1274590368,2549180736,3186475920,5098361472}
{1298435670,2596871340,3246089175,5193742680}
{1365079482,2730158964,3412698705,5460317928}
{1367590482,2735180964,3418976205,5470361928}
{1472590368,2945180736,3681475920,5890361472}
{1476859032,2953718064,3692147580,5907436128}
{1479365082,2958730164,3698412705,5917460328}
{1564329870,3128659740,3910824675,6257319480}
{1587296430,3174592860,3968241075,6349185720}
{1274590368,3186475920,3186475920,5098361472}
{1507936482,3015872964,3769841205,6031745928}
{1526437098,3052874196,3816092745,6105748392}
{1642958730,3285917460,4107396825,6571834920}
{1705483926,3410967852,4263709815,6821935704}
{1728439506,3456879012,4321098765,6913758024}
{1759048632,3518097264,4397621580,7036194528}
{1472590368,3681475920,3681475920,5890361472}
{1847356290,3694712580,4618390725,7389425160}
{1950637284,3901274568,4876593210,7802549136}
{1962530874,3925061748,4906327185,7850123496}
{1962730854,3925461708,4906827135,7850923416}
{1962850734,3925701468,4907126835,7851402936}
{1975308642,3950617284,4938271605,7901234568}
{2107653948,4215307896,5269134870,8430615792}
{2136507948,4273015896,5341269870,8546031792}
{2136759048,4273518096,5341897620,8547036192}
{2147936508,4295873016,5369841270,8591746032}
{2150793648,4301587296,5376984120,8603174592}
{2197530864,4395061728,5493827160,8790123456}
{2409513678,4819027356,6023784195,9638054712}

Paul.

• benvitalis says:

I had 29 solutions in mind, such that A + D is also pandigital.
You found 36. The seven extra solutions are also interesting since each of them is
10 times a pandigital number.

• benvitalis says:

{2107653948,4215307896,5269134870,8430615792}
2107653948+8430615792 = 10538269740

{2136507948,4273015896,5341269870,8546031792}
2136507948+8546031792 = 10682539740

{2136759048,4273518096,5341897620,8547036192}
2136759048+8547036192 = 10683795240

{2147936508,4295873016,5369841270,8591746032}
2147936508+8591746032 = 10739682540

{2150793648,4301587296,5376984120,8603174592}
2150793648+8603174592 = 10753968240

{2197530864,4395061728,5493827160,8790123456}
2197530864+8790123456 = 10987654320

{2409513678,4819027356,6023784195,9638054712}
2409513678+9638054712 = 12047568390

2. paul says:

Here are a few more that aren’t with a<b<c<d.

A + D = B and the arithmetic mean of A & D is = C

1098765432 5493827160 2746913580 4395061728
1269835704 6349178520 3174589260 5079342816
1270356984 6351784920 3175892460 5081427936
1274590368 6372951840 3186475920 5098361472
1472590368 7362951840 3681475920 5890361472
1476859032 7384295160 3692147580 5907436128
1098765432 7691358024 2746913580 4395061728
1098765432 8790123456 2746913580 4395061728
1759048632 8795243160 4397621580 7036194528
1950637284 9753186420 4876593210 7802549136
1234567890 9876543120 3086419725 4938271560
the geometric mean od A & D is

2197530864
2539671408
2540713968
2549180736
2945180736
2953718064
2197530864
2197530864
3518097264
3901274568
2469135780

Paul.

• benvitalis says: