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When I use the term ninedigital in these articles I always refer to a strictly zeroless pandigital (digits from 1 to 9 each appearing just once).

  Topic 3.12   [ September 15, 2003 ]
“Ninedigital fractions” equations and their distinct integer solutions
( sometimes generalized to “Ninedigital expressions” )

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by Terry Trotter (email)
by Jean Claude Rosa (email)
by Patrick De Geest (email)
by Jerry Levy (email)

Introduction see Levy Expressions by Terry Trotter (†)

Variation #0 : the original equation  A/BC + D/EF + G/HI  [L. Mittenzwey, circa 1880]

BC, EF and HI are not digit products but digit concatenations.

Only one solution

1₁

Go to Amazing Nine Digits Property

Variation #1 : the equation  A/(B*C) + D/(E*F) + G/(H*I)  [Jerry Levy's suggestion]

Four different values - 7 distinct solutions (see subscript values)

1₁, 2₂, 4₂, 5₂

Why do I underscore the specification distinct ? (info by J.C.R.)

We can say that, for example, the following 3 solutions are identical :
1/(3*6) + 5/(8*9) + 7/(2*4) = 1
1/(6*3) + 5/(8*9) + 7/(4*2) = 1
5/(9*8) + 1/(6*3) + 7/(2*4) = 1

Likewise for each solution we can produce an extra of 47 but identical ones,
meaning 48 in total (including the original solution). Why 48 ?
For each triplet, A/(B*C) for example, we have 2 possible dispositions :

A/(B*C) and A/(C*B)

There are 3 triplets so 8 possible dispositions ( 8 = 2*2*2 ).
On the other hand the order in which we write the triplets in the expression
has no effect on the final value.
Example:

N = 1/(3*6) + 5/(8*9) + 7/(2*4)

N = 5/(8*9) + 1/(3*6) + 7/(2*4)

There are 6 different dispositions among our 3 triplets so in total we have 6*8,
meaning 48 different dispositions giving the same result.

Samples
N = 1 ; 1/(3*6) + 5/(8*9) + 7/(2*4)

so 1 distinct solution.

N = 2 ; 5/(1*8) + 7/(2*4) + 9/(3*6)
N = 2 ; 5/(2*4) + 7/(1*8) + 9/(3*6)

so 2 distinct solutions.

N = 4 ; 5/(3*4) + 6/(8*9) + 7/(1*2)
N = 4 ; 8/(4*9) + 5/(3*6) + 7/(1*2)

so 2 distinct solutions.

N = 5 ; 5/(3*8) + 7/(4*6) + 9/(1*2)
N = 5 ; 5/(4*6) + 7/(3*8) + 9/(1*2)

so 2 distinct solutions.

Note that we have 7*48 or 336 undistinct solutions.

Variation #2 : the equation  (A/B)*C + (D/E)*F + (G/H)*I  [Jerry Levy's suggestion]

82 different values - 352 distinct solutions (see subscript values)

5₃, 6₂, 7₅, 8₃, 9₂,

10₅, 11₄, 13₄, 14₄, 15₄, 16₃, 17₇, 18₉, 19₇,

20₇, 21₁₃, 22₉, 23₅, 24₁₁, 25₈, 26₆, 27₄, 28₅, 29₃,

30₇, 31₂, 32₄, 34₆, 35₅, 36₉, 37₄, 38₃, 39₂,

40₅, 41₅, 42₈, 43₇, 44₃, 45₂, 47₃, 48₃, 49₁,

50₆, 52₃, 53₄, 54₇, 55₄, 56₃, 57₆, 58₇, 59₂,

60₃, 61₁, 62₄, 63₇, 64₄, 65₆, 66₄, 67₅, 68₆, 69₂,

70₃, 71₂, 72₃, 73₅, 74₆, 75₃, 76₂, 77₆, 78₂, 79₂,

80₂, 81₂, 82₁, 83₄, 84₃, 85₂, 89₂,

90₂, 91₁, 94₁, 96₂.

Ten numbers are not present between 5 and 96

12, 33, 46, 51, 86, 87, 88, 92, 93, 95

Variation #3 : the equation  (A/B)^C + (D/E)^F + (G/H)^I  [Jerry Levy's suggestion]

WTM column see Power-full Fractions by Terry Trotter

135 different values - 200 distinct solutions (see subscript values)

70₁,

132₁, 241₂, 356₁, 374₂, 407₁, 435₂, 593₂, 665₂, 672₆, 810₁, 882₁, 889₂,

1098₁, 1100₁, 2222₂, 2255₁, 2276₁, 2283₂, 2432₂, 2824₁, 2942₂, 2948₁, 3937₁, 4154₁,

4507₁, 6526₂, 6740₁, 6908₁, 7307₂, 7713₁, 7913₁, 9205₂, 9373₂, 9772₂, 9967₁,

10219₂, 14465₁, 15762₁, 16880₁, 17323₂, 17346₁, 17521₁, 17540₁, 17575₃, 17738₁,

17792₂, 17816₁, 17920₂, 17923₁, 18068₂, 18137₂, 19936₁, 20054₂, 20058₁, 20723₂,

20984₁, 21902₂, 22314₁, 22977₁, 23188₂, 23432₁, 23449₂, 24241₁, 32090₁, 33408₂,

33625₁, 35019₁, 35036₂, 36498₁, 59181₂, 59204₁, 59433₃, 61244₁, 67966₂, 75449₁,

78198₁, 78641₂, 78664₁, 78858₁, 78893₃, 79110₂, 82302₁, 84750₁, 84767₂, 97816₁,

117690₁, 117896₁, 118148₂, 118754₁, 124242₁, 262515₁, 262897₁, 263296₂,

264363₁, 264577₁, 278560₂, 278967₁, 279977₁, 280183₁, 280435₂, 281041₁,

286529₁, 340285₁, 391265₂, 391482₁, 392876₁, 392893₂,

1679987₁, 1681835₁, 1953257₂, 1953280₁, 1953509₃, 1955320₁, 1969525₁,

2097459₁, 2097476₂, 2097696₂, 2097913₁, 4783005₂, 4783028₁, 4783220₁,

4783257₃, 4784009₁, 5765108₁, 5765125₂, 5765345₂, 5765562₁,

40353643₂, 40353666₁, 40353858₁, 40353895₃, 40354647₁, 43046881₂,

134217888₂.

Six distinct solutions exist for integer value 672 due to sums of powers of 2 permutations.
Six cases with three distinct solutions : values 17575, 59433, 78893, 1953509, 4783257 & 40353895.

Samples

672 = (2/1)^5 + (6/3)^7 + (8/4)^9

672 = (2/1)^5 + (6/3)^9 + (8/4)^7

672 = (2/1)^7 + (6/3)^5 + (8/4)^9

672 = (2/1)^7 + (6/3)^9 + (8/4)^5

672 = (2/1)^9 + (6/3)^5 + (8/4)^7

672 = (2/1)^9 + (6/3)^7 + (8/4)^5

Fourteen numbers turned out to be prime.
Prime Curios! 241 is the smallest and Prime Curios! 4784009 is the largest of the lot.

Two numbers happen to be palindromic.
2222 is the first which is in fact also a repdigit.
23432 is the second one.

Samples

2222 = (6/2)^1 + (8/4)^5 + (9/3)^7

2222 = (6/2)^7 + (8/4)^5 + (9/3)^1

23432 = (4/2)^6 + (7/1)^5 + (9/3)^8

Variation #4 : the equation  A/(B+C) + D/(E+F) + G/(H+I)  [Jean Claude Rosa's suggestion]

Four different values - 87 distinct solutions (see subscript values)

1₃, 2₅₅, 3₁₈, 4₁₁

Samples

1 = 1/(2+4) + 5/(7+8) + 6/(3+9)

1 = 2/(6+9) + 3/(7+8) + 4/(1+5)

1 = 2/(7+8) + 3/(6+9) + 4/(1+5)

Variation #5 : the equation  (A+B)/C + (D+E)/F + (G+H)/I  [Jean Claude Rosa's suggestion]

24 different values - 355 distinct solutions (see subscript values)

3₆, 4₁₁, 5₁₉, 6₂₁, 7₁₂, 8₂₆, 9₂₃,

10₂₈, 11₁₆, 12₁₈, 13₁₃, 14₁₅, 15₁₄, 16₁₉, 17₁₁, 18₂₃, 19₁₆

20₁₉, 21₇, 22₁₃, 23₁₁, 24₇, 25₅, 26₂.

Samples

26 = (4+5)/3 + (6+8)/2 + (7+9)/1

26 = (4+6)/2 + (5+7)/3 + (8+9)/1

Note that 13 has 13 distinct solutions, ain't we lucky this time !!

Variation #6 : the equation  A/(B–C) + D/(E–F) + G/(H–I)  [Terry Trotter's suggestion]

45 different values - 8196 distinct solutions (see subscript values)

–24₆, –22₆, –20₁₄,

–19₄, –18₃₄, –17₆, –16₅₈, –15₁₈, –14₇₇, –13₃₀, –12₁₄₂, –11₅₈, –10₂₂₆,

–9₁₁₃, –8₃₀₅, –7₁₇₅, –6₃₉₀, –5₂₅₆, –4₅₀₆, –3₃₃₀, –2₆₂₈, –1₃₄₆,

0₇₄₀,

1₃₄₆, 2₆₂₈, 3₃₃₀, 4₅₀₆, 5₂₅₆, 6₃₉₀, 7₁₇₅, 8₃₀₅, 9₁₁₃,

10₂₂₆, 11₅₈, 12₁₄₂, 13₃₀, 14₇₇, 15₁₈, 16₅₈, 17₆, 18₃₄, 19₄,

20₁₄, 22₆, 24₆.

Samples

–19 = 4/(3–5) + 8/(1–2) + 9/(6–7)

–19 = 4/(3–5) + 8/(6–7) + 9/(1–2)

–19 = 6/(1–2) + 8/(5–7) + 9/(3–4)

–19 = 6/(3–4) + 8/(5–7) + 9/(1–2)

19 = 4/(5–3) + 8/(2–1) + 9/(7–6)

19 = 4/(5–3) + 8/(7–6) + 9/(2–1)

19 = 6/(2–1) + 8/(7–5) + 9/(4–3)

19 = 6/(4–3) + 8/(7–5) + 9/(2–1)

Note the symmetry in values and distinct solutions.

All values have composite distinct solutions, except for –9 and 9
which have a prime number of solutions namely 113.

Variation #7 : the equation  (A–B)/C + (D–E)/F + (G–H)/I  [Terry Trotter's suggestion]

17 different values - 1612 distinct solutions (see subscript values)

–8₅, –7₁₃, –6₃₅, –5₄₀, –4₁₁₈, –3₉₉, –2₂₁₂, –1₁₈₈,

0₁₉₂,

1₁₈₈, 2₂₁₂, 3₉₉, 4₁₁₈, 5₄₀, 6₃₅, 7₁₃, 8₅.

Values –2 and 2 each have the record palindromic number of solutions nl. 212.
We have prime number of solutions for the first and last two values.

Samples

–8 = (2–8)/1 + (4–9)/3 + (5–7)/6

–8 = (2–9)/1 + (3–6)/4 + (5–7)/8

–8 = (2–9)/1 + (4–8)/3 + (7–5)/6

–8 = (3–9)/1 + (4–7)/2 + (5–8)/6

–8 = (3–9)/1 + (4–7)/6 + (5–8)/2

8 = (5–7)/6 + (8–4)/3 + (9–2)/1

8 = (6–3)/4 + (7–5)/8 + (9–2)/1

8 = (7–4)/2 + (8–5)/6 + (9–3)/1

8 = (7–4)/6 + (8–5)/2 + (9–3)/1

8 = (7–5)/6 + (8–2)/1 + (9–4)/3

Variation #8 : the equation  (A+B)*C + (D+E)*F + (G+H)*I  [Terry Trotter's suggestion]

129 different values - 7560 distinct solutions (see subscript values)
This equation always deliver integer solutions whatever ninedigital combination is used.
In total we have factorial 9 (or 9! = 362880) undistinct solutions.
Values range from 70 to 198 without gaps.

70₁, 71₂, 72₅, 73₄, 74₆, 75₇, 76₉, 77₉, 78₁₇, 79₁₀,

80₁₁, 81₁₀, 82₁₂, 83₉, 84₉, 85₁₀, 86₁₁, 87₁₀, 88₁₄, 89₁₃,

90₁₅, 91₁₄, 92₂₂, 93₁₃, 94₂₁, 95₁₆, 96₂₂, 97₁₈, 98₃₂, 99₂₀,

100₃₆, 101₂₁, 102₄₂, 103₃₁, 104₄₁, 105₂₇, 106₄₈, 107₃₂, 108₆₄, 109₂₉,

110₆₁, 111₃₀, 112₆₁, 113₄₂, 114₇₃, 115₃₁, 116₅₇, 117₄₇, 118₇₈, 119₃₇,

120₇₄, 121₄₀, 122₈₃, 123₄₅, 124₆₈, 125₅₀, 126₁₀₁, 127₅₃, 128₉₇, 129₆₂,

130₉₄, 131₅₅, 132₁₄₄, 133₇₁, 134₁₀₂, 135₇₇, 136₁₀₂, 137₇₂, 138₁₅₆, 139₆₄,

140₁₀₂, 141₉₅, 142₁₃₄, 143₇₃, 144₁₃₆, 145₇₃, 146₁₁₁, 147₉₉, 148₁₃₅, 149₇₂,

150₁₅₉, 151₆₅, 152₁₄₀, 153₁₁₃, 154₁₂₄, 155₇₇, 156₁₆₁, 157₉₀, 158₁₅₃, 159₁₀₂,

160₁₃₉, 161₈₆, 162₂₀₆, 163₉₆, 164₁₁₇, 165₉₅, 166₁₃₀, 167₉₀, 168₁₈₀, 169₈₃,

170₁₁₈, 171₈₅, 172₁₁₃, 173₈₃, 174₁₂₅, 175₅₅, 176₇₆, 177₇₇, 178₇₈, 179₃₈,

180₅₆, 181₃₁, 182₅₈, 183₃₃, 184₃₆, 185₂₅, 186₃₄, 187₁₉, 188₂₅, 189₁₆,

190₁₈, 191₁₂, 192₁₂, 193₇, 194₈, 195₅, 196₄, 197₄, 198₃.

Value 162 yields the record number of distinct solutions nl. 206
Value 70 gives only 1 but unique distinct solution.

Samples

70 = (4+5)*3 + (6+7)*2 + (8+9)*1

198 = (1+2)*3 + (4+5)*6 + (7+8)*9

198 = (1+2)*3 + (4+5)*7 + (6+8)*9

198 = (1+2)*3 + (4+5)*8 + (6+7)*9

Variation #9 : the equation  (A+B)^C + (D+E)^F + (G+H)^I  [Terry Trotter's suggestion]

6283 different values - 7560 distinct solutions (see subscript values)
Here also all possible combinations yield integer results.
There are too many integer solutions to display them all. I shall
restrict myself to some subgroups and highlighting the curios.

Value 915 is the smallest solution.
Value 38443890843 is the largest solution.

16 values each have the record number of distinct solutions i.e. 6 :

2379₆, 1110000₆, 1932623₆, 2986140₆, 10101000₆, 19501823₆,

46941951₆, 100100100₆, 150999040₆, 214360223₆, 387486099₆,

387952659₆, 392203539₆, 430467219₆, 1000100010₆, 2357947823₆.

( Note the four binary like numbers ! )

Only 1 value has a unique number of distinct solutions namely 3 :

150706₃.

Samples

150706 = (1+7)^5 + (3+4)^6 + (8+9)^2

150706 = (1+8)^2 + (3+7)^5 + (6+9)^4

150706 = (1+9)^5 + (3+6)^2 + (7+8)^4

The following list shows how many values have exactly 1, 2, 3 and 6 distinct solutions :

V1₁ = 5071

V2₂ = 1195

V3₃ = 1

V6₆ = 16

Exact 10 palindromic values appear in the list.

969₂, 999₁, 6666₁, 30903₁, 47174₁,

67576₁, 761167₁, 844448₁, 1436341₁, 6055506₁.

The smallest palindrome 969 has two distinct solutions.

969 = (4+5)^3 + (6+9)^1 + (7+8)^2

969 = (4+5)^3 + (6+9)^2 + (7+8)^1

The beastly repdigit 6666 shows off !

6666 = (1+6)^4 + (5+8)^2 + (7+9)^3

Are we lucky ? since the first sum gives 1+6 = 7
Are we unlucky ? since the second sum gives 5+8 = 13
What about the third sum 7+9 = 16 ? Subtract its power 3... aargh !

Speaking of palindromes...
I count 287 primes with 1 distinct solution and
I count 76 primes with 2 distinct solutions.

Now note the next two palindromic results !!

287 + 76 = 363

287 * 76 = 21812

Exactly 1 tautonymic number appears in the list i.e. :

1515 = (4+7)^3 + (5+8)^2 + (6+9)^1 = (4+9)^2 + (5+6)^3 + (7+8)^1

Value 941 (Prime Curios!) is the smallest prime solution.
Value 38443418449 (Prime Curios!) is the largest prime solution.

This must be the ninedigital find of the year !
Look at the following unique dual ninedigital equation :

214358976 = (3 + 6)2 + (4 + 7)8 + (5 + 9)1

Variation #9 shows no other such ninedigital (or even pandigital) solutions.

Variation #10 : the equation  (A–B)*C + (D–E)*F + (G–H)*I  [Terry Trotter's suggestion]

205 different values - 60480 distinct solutions (see subscript values)
All possible combinations result in integer solution either positive or negative.
Values range from –102 to 102 without gaps.
The number of negative solutions is equal to the number of positive solutions.

0₈₇₀,

±1₄₅₃, ±2₈₄₁, ±3₄₄₅, ±4₇₂₁, ±5₄₈₇, ±6₇₉₉, ±7₄₉₃, ±8₇₅₆, ±9₄₇₄,

±10₇₈₆, ±11₄₃₂, ±12₇₄₄, ±13₄₄₆, ±14₇₃₇, ±15₄₆₀, ±16₆₈₇, ±17₄₃₆, ±18₇₃₃, ±19₄₀₁,

±20₆₉₄, ±21₃₉₂, ±22₆₆₀, ±23₄₃₇, ±24₅₆₃, ±25₃₇₄, ±26₆₃₆, ±27₄₀₂, ±28₅₆₄, ±29₃₆₂,

±30₆₁₀, ±31₃₀₉, ±32₅₄₀, ±33₃₀₇, ±34₄₉₆, ±35₃₅₈, ±36₅₀₄, ±37₂₉₂, ±38₄₆₈, ±39₂₆₂,

±40₄₃₄, ±41₂₈₉, ±42₄₄₀, ±43₂₅₉, ±44₄₂₀, ±45₂₆₄, ±46₃₇₅, ±47₂₃₈, ±48₃₄₇, ±49₂₂₆,

±50₃₈₂, ±51₁₉₀, ±52₃₀₆, ±53₁₇₆, ±54₃₂₃, ±55₁₉₃, ±56₂₆₆, ±57₁₅₂, ±58₂₆₀, ±59₁₄₇,

±60₂₃₀, ±61₁₄₅, ±62₂₂₃, ±63₁₃₃, ±64₁₇₈, ±65₁₂₈, ±66₁₇₂, ±67₁₀₁, ±68₁₆₅, ±69₁₀₁,

±70₁₄₇, ±71₉₃, ±72₁₅₀, ±73₈₅, ±74₁₃₅, ±75₇₉, ±76₁₁₄, ±77₇₃, ±78₁₀₃, ±79₅₈,

±80₈₈, ±81₆₂, ±82₉₂, ±83₅₂, ±84₈₂, ±85₄₈, ±86₆₂, ±87₅₀, ±88₅₉, ±89₄₂,

±90₇₅, ±91₃₁, ±92₄₁, ±93₃₄, ±94₃₃, ±95₂₅, ±96₂₅, ±97₁₃, ±98₁₁, ±99₉,

±100₇, ±101₂, ±102₁.

32 values have palindromic distinct solutions :

±9, ±14, ±26, ±37, ±39, ±40, ±54, ±67, ±69, ±80, ±94, ±98, ±99, ±100, ±101, ±102.

16 values can be grouped in four quartets each having the same number of distinct solutions :

[±95 & ±96]₂₅, [±81 & ±86]₆₂, [±67 & ±69]₁₀₁, [±59 & ±70]₁₄₇

40 values have prime distinct solutions :

±5, ±18, ±19, ±24, ±33, ±48, ±55, ±62, ±67, ±69,

±75, ±77, ±78, ±88, ±91, ±92, ±97, ±98, ±100, ±101

2 values only shows one unique distinct solution :

±102₁

Samples

0 = (1–2)*3 + (4–5)*6 + (8–7)*9

0 = (9–8)*7 + (6–5)*2 + (1–4)*3

when the value is zero we have in fact pandigital expressions !

–101 = (1–8)*9 + (2–7)*6 + (3–5)*4

–101 = (1–9)*8 + (2–7)*5 + (3–6)*4

101 = (5–3)*4 + (7–2)*6 + (8–1)*9

101 = (6–3)*4 + (7–2)*5 + (9–1)*8

–102 = (1–9)*8 + (2–7)*6 + (3–5)*4

102 = (5–3)*4 + (7–2)*6 + (9–1)*8

Variation #11 : the equation  (A–B)^C + (D–E)^F + (G–H)^I  [Terry Trotter's suggestion]

10426 different values - 60480 distinct solutions (see subscript values)
All possible combinations result in integer solution either positive or negative.
Like variation #9 there are too many integer solutions to display. Therefore I shall
restrict myself to some interesting categories and/or highlighting the curios.
Values range from –40369992 to 40370007 but not consecutively, there are gaps.

3649 values are negative, 1 value is equal to 0 and 6776 values are positive.
Note that this last total is palindromic !
Variation #11 shows for the first time that the negative values do not match the positive values.

The largest prime is 40356733 (out of 733 see Prime Curios!).

40356733 = (3–4)^6 + (7–2)^5 + (8–1)^9

40356733 = (4–3)^6 + (7–2)^5 + (8–1)^9

The smallest negative palindrome is –97079
The largest palindrome (out of 179) is 5764675
Samples

Value of the beast ! 12 distinct solutions ( note that digit 6 is always 'in' power ! ) 666 = (1–2)^8 + (4–7)^6 + (5–9)^3 666 = (1–2)^8 + (5–9)^3 + (7–4)^6 666 = (1–4)^6 + (5–9)^3 + (7–8)^2 666 = (1–4)^6 + (5–9)^3 + (8–7)^2 666 = (1–5)^3 + (4–7)^6 + (8–9)^2 666 = (1–5)^3 + (4–7)^6 + (9–8)^2 666 = (1–5)^3 + (7–4)^6 + (8–9)^2 666 = (1–5)^3 + (7–4)^6 + (9–8)^2 666 = (2–1)^8 + (4–7)^6 + (5–9)^3 666 = (2–1)^8 + (5–9)^3 + (7–4)^6 666 = (4–1)^6 + (5–9)^3 + (7–8)^2 666 = (4–1)^6 + (5–9)^3 + (8–7)^2 The beast enclosed 46664 = (1–4)^2 + (3–9)^6 + (7–8)^5 46664 = ... (46664 has like 666 also 12 distinct solutions !) See also WONplate 154

Some repdigits –7777 = (2–4)^1 + (3–9)^5 + (6–7)^8 –7777 = (2–4)^1 + (3–9)^5 + (7–6)^8 –7777 = (2–4)^1 + (3–9)^5 + (7–8)^6 –7777 = (2–4)^1 + (3–9)^5 + (8–7)^6 –7777 = (2–8)^5 + (3–4)^6 + (7–9)^1 –7777 = (2–8)^5 + (4–3)^6 + (7–9)^1 –7777 = (3–9)^5 + (4–6)^1 + (7–8)^2 –7777 = (3–9)^5 + (4–6)^1 + (8–7)^2 1111 = (2–6)^4 + (7–5)^9 + (8–1)^3 1111 = (6–2)^4 + (7–5)^9 + (8–1)^3 1111 = (6–9)^4 + (7–3)^5 + (8–2)^1 1111 = (7–3)^5 + (8–2)^1 + (9–6)^4 7777 = (4–7)^1 + (6–8)^2 + (9–3)^5 7777 = (4–7)^1 + (8–6)^2 + (9–3)^5

Exactly 4 negative and 12 positive tautonymic numbers can be retrieved from the list :

–7272₂, –3636₄, –3131₂, –1919₄

1313₁₆, 2020₄, 2323₂, 2525₄, 2828₄, 3131₅, 3434₄, 3636₄, 3939₂, 6565₁₆, 6868₈, 7272₂

Two of them (see underscored) are used in Zager and Evans' song 'In the Year 2525'.

There is one value having the record number of distinct solutions namely... 1 itself !!

1₅₄₄

Smallest : 1 = (1–2)^3 + (4–5)^6 + (8–7)^9

Largest  : 1 = (9–8)^7 + (6–5)^4 + (2–3)^1

We can distinguish 106 pandigital expressions with distinct solutions.

0₁₀₆

Smallest : 0 = (1–3)^8 + (2–6)^4 + (5–7)^9

Largest  : 0 = (9–8)^7 + (6–5)^3 + (2–4)^1

87 different distinct solutions can be found for the various different values.

( Note : d.s. = total distinct solutions ).

1 - 3556 values have 2 d.s. 2 - 2882 values have 4 d.s. 3 - 1124 values have 1 d.s. 4 - 1054 values have 8 d.s. 5 - 436 values have 6 d.s. 6 - 237 values have 12 d.s. 7 - 197 values have 16 d.s. 8 - 174 values have 10 d.s. 9 - 93 values have 3 d.s. 10 - 70 values have 20 d.s. 11 - 68 values have 24 d.s. 12 - 63 values have 5 d.s. 13 - 47 values have 14 d.s. 14 - 44 values have 18 d.s. 15 - 43 values have 32 d.s. 16 - 29 values have 28 d.s. 17 - 22 values have 9 d.s. 18 - 18 values have 36 d.s. 19 - 15 values have 7 d.s. 20 - 15 values have 22 d.s. 21 - 14 values have 11 d.s. 22 - 13 values have 40 d.s. 23 - 12 values have 44 d.s. 24 - 11 values have 26 d.s. 25 - 11 values have 38 d.s. 26 - 10 values have 30 d.s. 27 - 9 values have 13 d.s. 28 - 9 values have 34 d.s. 29 - 9 values have 48 d.s. 30 - 9 values have 64 d.s. 31 - 8 values have 15 d.s. 32 - 8 values have 21 d.s. 33 - 7 values have 46 d.s. 34 - 6 values have 35 d.s. 35 - 6 values have 42 d.s. 36 - 6 values have 60 d.s. 37 - 5 values have 17 d.s. 38 - 5 values have 52 d.s. 39 - 5 values have 58 d.s.

40 - 4 values have 50 d.s. (–243, –126, 72, 88) 41 - 4 values have 56 d.s. (–25, 31, 46658, 262144) 42 - 4 values have 68 d.s. (10, 23, 33, 1296) 43 - 4 values have 72 d.s. (–27, 264, 6561, 65536) 44 - 4 values have 76 d.s. (–4, 29, 129, 273) 45 - 3 values have 49 d.s. (28, 84, 144) 46 - 3 values have 54 d.s. (–125, 15, 35) 47 - 3 values have 92 d.s. (21, 38, 321) 48 - 2 values have 29 d.s. (94, 98) 49 - 2 values have 78 d.s. (13, 69) 50 - 2 values have 79 d.s. (–6, 2) 51 - 2 values have 84 d.s. (25, 127) 52 - 2 values have 90 d.s. (–5, 261) 53 - 2 values have 94 d.s. (–510, 262146) 54 - 2 values have 106 d.s. (0, 514) 55 - 1 value has 25 d.s. (52) 56 - 1 value has 31 d.s. (228) 57 - 1 value has 39 d.s. (–12) 58 - 1 value has 41 d.s. (26) 59 - 1 value has 47 d.s. (12) 60 - 1 value has 53 d.s. (218) 61 - 1 value has 59 d.s. (114) 62 - 1 value has 62 d.s. (–262142) 63 - 1 value has 66 d.s. (253) 64 - 1 value has 74 d.s. (36) 65 - 1 value has 82 d.s. (22) 66 - 1 value has 91 d.s. (–2) 67 - 1 value has 102 d.s. (8) 68 - 1 value has 104 d.s. (27) 69 - 1 value has 108 d.s. (11) 70 - 1 value has 120 d.s. (4096) 71 - 1 value has 128 d.s. (4098) 72 - 1 value has 132 d.s. (–3) 73 - 1 value has 138 d.s. (81) 74 - 1 value has 150 d.s. (9) 75 - 1 value has 168 d.s. (5) 76 - 1 value has 170 d.s. (16) 77 - 1 value has 171 d.s. (6) 78 - 1 value has 176 d.s. (64) 79 - 1 value has 177 d.s. (256) 80 - 1 value has 180 d.s. (4) 81 - 1 value has 181 d.s. (66) 82 - 1 value has 182 d.s. (7) 83 - 1 value has 189 d.s. (258) 84 - 1 value has 202 d.s. (18) 85 - 1 value has 348 d.s. (3) 86 - 1 value has 350 d.s. (–1) 87 - 1 value has 544 d.s. (1)

Variation #12 : the equation  A^B/C + D^E/F + G^H/I  [Jerry Levy's suggestion]

1428 different values - 1492 distinct solutions (see subscript values)
Values range from 40 to 134311552 but not consecutively, there are gaps.
Samples

40 = 3^5/9 + 6^1/8 + 7^2/4

134311552 = 4^5/2 + 6^7/3 + 8^9/1

The smallest prime is 317 (out of 168).
The largest prime is 67202801.
Samples

317 = 2^7/1 + 3^5/9 + 6^4/8

67202801 = 5^4/1 + 6^7/3 + 8^9/2

Almost all values (1364) have just 1 distinct solution.
Only 64 values show 2 distinct solutions.

6 values turn out to be palindromic.

787₁, 2112₁, 12621₁, 33133₁, 37673₁ and 39793₁.

Only the first one is prime as well.
Semiprime 37673 equals the product of two palprimes nl. 101 * 373 .

Exactly 2 tautonymic numbers can be extracted from the list :

4343 = 2^5/8 + 3^7/9 + 4^6/1

875875 = 4^7/1 + 6^8/2 + 9^5/3

Variation #13 : the equation  A^(B/C) + D^(E/F) + G^(H/I)  [Jerry Levy's suggestion]

259 different values - 468 distinct solutions (see subscript values)
Values range from 30 to 134217881 but not consecutively, there are gaps.

The smallest prime is 53 (out of 26).
The largest prime is 5764907.
Samples

53 = 1^(9/7) + 3^(6/2) + 5^(8/4)

5764907 = 5^(4/2) + 7^(8/1) + 9^(6/3)

10 palindromes can be perceived :

77, 131, 242, 282, 353, 404, 545, 707, 969, 33033

The sixth one [ 404 ] is not a HTML error !

7 different distinct solutions can be found for the various different values.
( Note : d.s. = total distinct solutions ).

1 - 135 values have 1 d.s. 2 - 96 values have 2 d.s. 3 - 14 values have 4 d.s. 4 - 8 values have 6 d.s. 5 - 3 values have 8 d.s. 6 - 2 values have 3 d.s. 7 - 1 value has 7 d.s.

Note that 1 nice value shows a unique property of having exactly 7 distinct solutions.

369₇

Samples

369 = 1^(2/6) + 5^(8/4) + 7^(9/3)

369 = 1^(3/9) + 5^(8/4) + 7^(6/2)

369 = 1^(6/2) + 5^(8/4) + 7^(9/3)

369 = 1^(6/8) + 5^(4/2) + 7^(9/3)

369 = 1^(8/6) + 5^(4/2) + 7^(9/3)

369 = 1^(9/3) + 5^(8/4) + 7^(6/2)

369 = 2^(7/1) + 5^(8/4) + 6^(9/3)

Variation #14 : the equation  (A+B)*C + (D–E)/F + G^H*I 
[Terry Trotter's all five operations suggestion]

13253 different values - 45360 distinct solutions (see subscript values)
Values range from 24 to 939524151 but not consecutively, there are gaps.

Let me quote Terry himself about the expression (A + B)*C + (D – E)/F + G^H*I :

“ I have designed this new variant for our Levy Expressions project -- #14.
Since I did add it to my webpage, I thought you might like to know about it, at least.
Note that all the previous expressions used but 2 operations in each triad.
Now I'm using all five operations, based on the following logic :
We (or at least I did) teach in school math the following hierarchy of operations :

1. addition

2. subtraction

3. multiplication

4. division

5. exponentiation

This goes from easy to 'hard', for most kids. At least, we teach them in that order.
Plus when we study logarithms, we show that to multiply two numbers, we add the logs;
when we divide two numbers, we subtract their logs; and when we need to raise a number
to a power, i.e. exponentiate, we multiply the base's log by the exponent. Hence,
I used #1 & 3 with A, B & C; used #2 & 4 with D, E & F; and used #3 & 5 with G, H & I.
Hey, it may be corny, but I'm trying to teach kids to like math & look at it in a new way. :>)
And it does review the order of operations with a novel approach. Hope you like it. ”

The first thing that catches my attention is that this variation produced a record number of
different values nl. 13253. This can makes room for lots of discoveries I hope.

This year 2003 for instance can be expressed in 4 distinct ways :

2003 = (2+5)*8 + (7–4)/1 + 6^3*9

2003 = (2+7)*6 + (9–4)/1 + 3^5*8

2003 = (4+7)*6 + (2–9)/1 + 3^5*8

2003 = (6+7)*4 + (9–2)/1 + 3^5*8

Values 66, 666 and 6666 are part of the lot !
Samples

66= (2+5)*8 + (7–3)/4 + 1^6*9

666 = (4+7)*8 + (5–3)/1 + 2^6*9

6666 = (6+7)*8 + (5–2)/3 + 9^4*1

The smallest prime is 29 (out of 1198).
The largest prime is 939524147.
Samples

29 = (3+7)*2 + (9–4)/5 + 1^6*8

939524147 = (4+6)*5 + (3–1)/2 + 8^9*7

152 palindromes can be detected.
The largest one is 327723 = (6+7)*3 + (9–1)/2 + 4^8*5

This variation has also the record largest tautonymic number i.e. 19131913₂ !

1913_1913 = (1+6)*5 + (8–2)/3 + 9^7*4

1913_1913 = (2+5)*6 + (3–8)/1 + 9^7*4

Other smaller samples

118118₁ = (5+7)*2 + (4–8)/1 + 3^9*6

177177₁ = (6+7)*2 + (8–4)/1 + 9^5*3

413413₁ = (3+6)*8 + (2–4)/1 + 9^5*7

839839₃ = (2+4)*5 + (9–1)/8 + 6^7*3 = (2+4)*5 + (9–8)/1 + 6^7*3 = (5+8)*2 + (9–4)/1 + 6^7*3

  Topic 3.11   [ August 31, 2003 ]
Pandigitals divisible by 5-digit palprimes
by Jean Claude Rosa (email)

Jean Claude continued “WONplate 114 continued...” and researched the case of
pandigitals divisible by a palindromic prime or palprime for short and more in particular
by a 5-digit palprime (in total there are 93 - from 10301 to 98689).
24 palprimes are concerned in 27 solutions. No results exist for the 'not listed' palprimes !

number of solutions 1 for the following 5-digit palprimes :

12421, 12721, 15451, 16661, 30203, 30403, 30803,

32323, 36563, 37573, 38783, 71317, 73637, 76367,

76667, 77477, 90709, 93239, 93739, 97379, 97579

smallest palprime: 1037625498/12421 = 83538
largest palprime: 8713609542/97579 = 89298

number of solutions 2 for the following 5-digit palprimes :

14741, 19891, 34543

7829461035/14741 = 531135

8106739245/14741 = 549945

1253849076/19891 = 63036

1432689057/19891 = 72027

3140269587/34543 = 90909

5604981723/34543 = 162261

There are some curious palindromes expressable in two ways.

1037625498/12421 = 83538 = 1290745638/15451

1670492835/30403 = 54945 = 1692470835/30803

3504892617/36563 = 95859 = 8695274031/90709

2938451607/32323 = 90909 = 3140269587/34543

And of course the obligatory Number of the Beast in

8257091634/16661 = 495594

3709456128/76667 = 48384

Perhaps someone would like to extend the ninedigital variant of this same exercise.

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  Topic 3.10   [ August 25, 2003 ]
WONplate 114 continued...
by Jean Claude Rosa (email)

WONPlate 114 discusses Palindromic quotients through pandigital divisions
and only the denominators 2 to 9 were considered. Jean Claude extended the topic
and investigated the pandigitals divided by palindromic primes limiting himself to 11
and the fifteen threedigit palprimes from 101 to 929.

Divisor 11 gave 14 solutions.

divisor 11
number of solutions: 14
smallest: 2063784591/11 = 187616781
largest: 8459120637/11 = 769010967

This number of solutions (#14) already predicts how many of the fifteen palprimes will yield results !

divisor 101
number of solutions: nihil

The fourteen remaining palprimes give the following data

divisor 131 number of solutions: 4 smallest: 1023658497/131 = 7814187 largest: 6213785094/131 = 47433474 divisor 151 number of solutions: 3 smallest: 1374086259/151 = 9099909 largest: 6509781234/151 = 43111134 divisor 181 number of solutions: 3 smallest: 1078324695/181 = 5957595 largest: 1647083529/181 = 9099909 divisor 191 number of solutions: 7 smallest: 1304985726/191 = 6832386 largest: 9781342065/191 = 51211215 divisor 313 number of solutions: 4 smallest: 1078423659/313 = 3445443 largest: 2914386507/313 = 9311139 divisor 353 number of solutions: 3 smallest: 2108749635/353 = 5973795 largest: 4196750283/353 = 11888811 divisor 373 number of solutions: 6 smallest: 1028947356/373 = 2758572 largest: 7219560843/373 = 19355391

divisor 383 number of solutions: 3 smallest: 1263487509/383 = 3298923 largest: 2106478935/383 = 5499945 divisor 727 number of solutions: 4 smallest: 4069713285/727 = 5597955 largest: 6870549123/727 = 9450549 divisor 757 number of solutions: 5 smallest: 1982705634/757 = 2619162 largest: 5481760239/757 = 7241427 divisor 787 number of solutions: 4 smallest: 2389570461/787 = 3036303 largest: 7129046583/787 = 9058509 divisor 797 number of solutions: 4 smallest: 1930526874/797 = 2422242 largest: 5360841972/797 = 6726276 divisor 919 number of solutions: 6 smallest: 1587362049/919 = 1727271 largest: 6971402583/919 = 7585857 divisor 929 number of solutions: 5 smallest: 1765082349/929 = 1899981 largest: 8095471362/929 = 8714178

In total we have 61 solutions, a nice prime !

There are some curious palindromes expressable in two ways.

1746082539/919 = 1899981 = 1765082349/929

1374086259/151 = 9099909 = 1647083529/181

As always the Number of the Beast lurks around the corner
(two corners to be more precise!).

5048937162/757 = 6669666

Perhaps someone would like to extend the ninedigital variant of this same exercise
and see if there's anything interesting going on there, like for instance
the following unique nine- & pandigital equation :

832591674/181 = 4599954 = 1439785602/313

ps. the Number of the Ninedigital Beast houses here :

598231674/131 = 4566654

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  Topic 3.9   [ August 20, 2003 ]
Pandigital divisions equal to ratio N/D

Statistics : abcd = N/D * efghi   or   abcde = N/D * fghij

Terry Trotter (email) inspired by Puzzle 121 from Michael Winckler's site (Puzzle No. 121),
expanded the topic by investigating more ratios N/D and looking not only at ninedigit numbers

e.g. 5823 / 17469 = 1 / 3

e.g. 9876 / 12345 = 4 / 5

but also at the 10-digit pandigital version.

e.g. 13485 / 26970 = 1 / 2

e.g. 71496 / 20853 = 7 / 24

The ninedigit version with ratios 1/2, 1/3 up to 1/9 are already presented in WONplate 107
Now Terry has data for all the fractions in the range of 0 < N < D < 20
for both 9-digit & 10-digit work.
For this purpose he asked and got a ubasic program written by Carlos Rivera.
The code is short and straightforward so I'll list it here (some minor modifications from myself pdg).
(Change in line 20 the values N1 and N2 for other fractions)

   10   'trotter.ub
   20   B1=1000:B2=99999:N1=4:N2=5:Cc=0
   30   FF=6469693230                             : 'use FF=3234846615 for 9-digit version
   35   ' FF=1:for I=0 to 9:FF*=prm(I+1):next I   : 'use for I=1 to 9 for 9-digit version
   40   for B=B1 to B2:if B@N2<>0 then goto 70
   50   A=B*N1\N2:Z=A*10^alen(B)+B
   60   gosub *TEST:if T=1 then inc Cc:print Cc,A;"=";N1;"/";N2;"*";B
   70   next B
   80   end
   90   *TEST:T=0:F=1:L=alen(Z)
  100   for I=1 to L:D=val(mid(str(Z),I+1,1)):P=prm(D+1)
  110   if F@P=0 then cancel for:return
  120   F*=P:next I
  130   if F=FF then T=1
  140   return

and finally the statistics, the program output :

Part 1.

All possible ninedigit solutions (numerator > 1) and (denominator < 10)

2 / 5 = 6894 / 17235 2 / 5 = 8694 / 21735 2 / 5 = 9486 / 23715   3 solutions 4 / 5 = 9876 / 12345   1 unique and beautiful solution 2 / 7 = 3654 / 12789 2 / 7 = 3674 / 12859 2 / 7 = 5342 / 18697 2 / 7 = 7418 / 25963 2 / 7 = 9786 / 34251 2 / 7 = 9862 / 34517   6 solutions 2 / 9 = 3924 / 17658 2 / 9 = 7596 / 34182   2 solutions

(numerator ⩾ 1) and (10 < denominator < 20)

2 / 11 = 4716 / 25938 2 / 11 = 9432 / 51876 2 / 11 = 9486 / 52173   3 solutions 4 / 11 = 6492 / 17853 4 / 11 = 7836 / 21549   2 solutions 5 / 11 = 9765 / 21483   1 unique solution 1 / 12 = 3816 / 45792 1 / 12 = 6129 / 73548 1 / 12 = 7461 / 89532 1 / 12 = 7632 / 91584   4 solutions 5 / 12 = 8235 / 19764   1 unique solution 1 / 13 = 5184 / 67392 1 / 13 = 6273 / 81549 1 / 13 = 7281 / 94653   3 solutions 2 / 13 = 2538 / 16497 2 / 13 = 3876 / 25194 2 / 13 = 3918 / 25467 2 / 13 = 4518 / 29367 2 / 13 = 6972 / 45318 2 / 13 = 7896 / 51324 2 / 13 = 8196 / 53274   7 solutions 5 / 13 = 4765 / 12389 5 / 13 = 8345 / 21697 5 / 13 = 8365 / 21749   3 solutions 1 / 14 = 1839 / 25746 1 / 14 = 1956 / 27384   my birthyear! - pdg 1 / 14 = 2967 / 41538 1 / 14 = 3297 / 46158 1 / 14 = 3678 / 51492 1 / 14 = 3912 / 54768 1 / 14 = 4398 / 61572 1 / 14 = 4713 / 65982   8 solutions 5 / 14 = 4635 / 12978   1 unique solution 1 / 15 = 1863 / 27945 1 / 15 = 6183 / 92745   2 solutions 8 / 15 = 9432 / 17685   1 unique solution 1 / 16 = 2871 / 45936 1 / 16 = 4581 / 73296 1 / 16 = 6147 / 98352   3 solutions 3 / 16 = 3294 / 17568 3 / 16 = 3672 / 19584 3 / 16 = 5967 / 31824   3 solutions 5 / 16 = 9765 / 31248   1 unique solution 7 / 16 = 5796 / 13248 7 / 16 = 9387 / 21456   2 solutions 11 / 16 = 9867 / 14352   1 unique solution

1 / 17 = 1579 / 26843 1 / 17 = 1679 / 28543 1 / 17 = 1738 / 29546 1 / 17 = 2174 / 36958 1 / 17 = 2689 / 45713 1 / 17 = 2693 / 45781 1 / 17 = 3217 / 54689 1 / 17 = 3478 / 59126 1 / 17 = 3821 / 64957 1 / 17 = 3841 / 65297 1 / 17 = 3952 / 67184 1 / 17 = 3954 / 67218 1 / 17 = 4519 / 76823 1 / 17 = 4523 / 76891 1 / 17 = 4596 / 78132 1 / 17 = 4619 / 78523 1 / 17 = 4623 / 78591 1 / 17 = 4796 / 81532 1 / 17 = 4916 / 83572 1 / 17 = 4921 / 83657 1 / 17 = 5261 / 89437 1 / 17 = 5263 / 89471 1 / 17 = 5273 / 89641 1 / 17 = 5378 / 91426 1 / 17 = 5461 / 92837 1 / 17 = 5463 / 92871 1 / 17 = 5478 / 93126   27 solutions 2 / 17 = 8514 / 72369   1 unique solution 3 / 17 = 5184 / 29376   1 unique solution 4 / 17 = 4536 / 19278 4 / 17 = 7428 / 31569   2 solutions 5 / 17 = 6435 / 21879 5 / 17 = 8145 / 27693   2 solutions 1 / 18 = 1593 / 28674   1 unique solution 7 / 18 = 8379 / 21546   1 unique solution 1 / 19 = 2736 / 51984 1 / 19 = 4293 / 81567   2 solutions 2 / 19 = 5136 / 48792 2 / 19 = 5238 / 49761 2 / 19 = 7254 / 68913 2 / 19 = 9132 / 86754   4 solutions 3 / 19 = 4671 / 29583   1 unique solution 4 / 19 = 3492 / 16587   1 unique solution 5 / 19 = 9165 / 34827   1 unique solution 6 / 19 = 4968 / 15732 6 / 19 = 8694 / 27531   2 solutions 7 / 19 = 7938 / 21546   1 unique solution 8 / 19 = 5368 / 12749 8 / 19 = 5864 / 13927 8 / 19 = 5872 / 13946   3 solutions

Part 2.

(numerator ⩾ 1) and (denominator < 20)
Listing all possible pandigital solutions would bring us too far.
Instead I opt for displaying only the 20 unique pandigital solutions.
With the aid of the above ubasic program one should easily find all the other solutions.

1 / 7 = 14076 / 98532 5 / 9 = 15930 / 28674 8 / 9 = 47016 / 52893 6 / 11 = 27486 / 50391 7 / 12 = 17829 / 30564 7 / 13 = 32571 / 60489 14 / 15 = 86310 / 92475 5 / 16 = 23490 / 75168 6 / 17 = 19872 / 56304 8 / 17 = 46152 / 98073

12 / 17 = 51084 / 72369 15 / 17 = 46710 / 52938 16 / 17 = 57312 / 60894 11 / 18 = 48312 / 79056 17 / 18 = 19584 / 20736 2 / 19 = 10248 / 97356 4 / 19 = 17460 / 82935 6 / 19 = 25704 / 81396 13 / 19 = 45981 / 67203 14 / 19 = 49518 / 67203

The top 5 pandigital fractions

4 / 5   78 solutions

1 / 2   48 solutions

10 / 17   27 solutions

5 / 13   20 solutions

2 / 7   19 solutions

Part 3.

Arriving at this point, it is time to relax from the serious work.
As there is no fractional limit (?) one can investigate beyond the denominator
limit that Terry imposed upon himself and go beyond 20 to discover more curios and trivia.
Again, feel free to use the above ubasic program.

13 and 666 are well-known numbers for their negative connotation
(the unlucky number and the number of the beast).
There exist now a single fraction such that brings these two numbers together :

13 / 666 = 1287 / 65934

ps. 65934 – 1287 = 64647   almost a palindrome, ain't I unlucky ?!
Also no such luck for the pandigital version with that same fraction.
The number of the beast must like this number 65934 as I found
the following beastly combinatorial equations.

666 / 13 = 65934 / 1287 666 / 18 = 65934 / 1782 666 / 22 = 65934 / 2178 666 / 29 = 65934 / 2871 666 / 72 = 65934 / 7128 666 / 79 = 65934 / 7821 666 / 83 = 65934 / 8217 666 / 88 = 65934 / 8712

666 / 122 = 65934 / 12078 666 / 130 = 65934 / 12870 666 / 172 = 65934 / 17028 666 / 180 = 65934 / 17820 666 / 213 = 65934 / 21087 666 / 220 = 65934 / 21780 666 / 283 = 65934 / 28017 666 / 290 = 65934 / 28710 666 / 718 = 65934 / 71082 666 / 720 = 65934 / 71280 666 / 788 = 65934 / 78012 666 / 790 = 65934 / 78210 666 / 829 = 65934 / 82071 666 / 830 = 65934 / 82170 666 / 879 = 65934 / 87021 666 / 880 = 65934 / 87120

Other numbers than 666 can be linked with "other than 65934" numerators.
Do there exist such numbers with more combinatorial solutions than the above 8/16 ?

ps. Add the 8 denominators (13+18+...) from the ninedigital version together.
An interesting palindrome arises, isn't it ! See WONplate 104

Part 4.

Ninedigital Fractions equal to Pandigital Fractions

Some 'unique' solutions.

5 / 16 = 9765 / 31248 = 23490 / 75168

4 / 19 = 3492 / 16587 = 17460 / 82935

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  Topic 3.8   [ December 25, 2001 ]
Ninedigital Smith numbers

Smith numbers are composites such that
'the sum of their digits' equal 'the sum of the digits of their prime factors'.
Shyam Sunder Gupta, editor of the site Number Recreations
investigated the ninedigital versions and found that

the smallest solution is 123456879 = 3 * 3 * 3 * 3 * 3 * 3 * 7 * 13 * 1861.

since 1 + 2 + 3 + 4 + 5 + 6 + 8 + 7 + 9 = 45 = 3 + 3 + 3 + 3 + 3 + 3 + 7 + 1 + 3 + 1 + 8 + 6 + 1

the largest solution is 987653214 = 2 * 3 * 3 * 7151 * 7673.

since 9 + 8 + 7 + 6 + 5 + 3 + 2 + 1 + 4 = 45 = 2 + 3 + 3 + 7 + 1 + 5 + 1 + 7 + 6 + 7 + 3

Source : Prime Curios!

Ninedigital see 123456879 and 987653214

Pandigital see 1023465798 and ?? Can You Find ??

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  Topic 3.7   [ February 10, 2001 ]
Pandigital pointing to Ninedigital

The 1234567890^th palindrome is 234567891198765432

This last palindrome is a ninedigital number 234567891 combined with its reversal 198765432.
Eric Schmidt developed a nice Java Applet where you can enter any
positive integer N and after pressing ENTER calculates the N^th palindrome.

Positive Integer Palindromes

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  Topic 3.6   [ January 13, 2001 ]
Amazing Nine Digits Property
from sci.math by Preon

(a/bc) + (d/ef) + (g/hi) = 1

where a through i are the unique digits 1 through 9.
bc, ef and hi are the concatenations (not the products) of b with c, e with f and h with i.
The puzzle seems to have only one solution... as I discovered with a search !

The solution is very easy to search for, Preon tells. It is

(9/12) + (7/68) + (5/34) = 1

" The solution is unique as my program revealed, took maybe 10 minutes
to write the code and 1 minute to run and display the only answer."

Preon wonders if someone has a proof or explanation
cause he hasn't a clue why this amazing property should be true...

Paul E. Triulzi, a senior IT specialist (email), noted that the equation
from Preon has some interesting symmetry [ August 31, 2001 ].

1. The numerators are all odd and sequential.
2. The numerator and denominator numbers are sequential
   (9,1,2 wraps around if 0 is not counted).

Preon investigated also the following pandigital variant
but found only solutions with the zero as a nonsignificant leading zero !

(a/bc) + (d/ef) + (g/hi) = j

(3÷04) + (6÷78) + (9÷52) = 1 (3÷48) + (7÷16) + (9÷02) = 5 (4÷28) + (5÷63) + (7÷09) = 1 (4÷07) + (5÷28) + (9÷36) = 1 (4÷28) + (5÷07) + (9÷63) = 1 (4÷06) + (7÷39) + (8÷52) = 1 (4÷36) + (7÷18) + (9÷02) = 5 (5÷02) + (6÷14) + (7÷98) = 3 (5÷38) + (6÷04) + (7÷19) = 2 (5÷13) + (6÷04) + (9÷78) = 2 (5÷09) + (7÷63) + (8÷24) = 1 (5÷36) + (7÷04) + (9÷81) = 2

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  Topic 3.5  
Extra pan- & ninedigital equations some smallest solutions for each type
by G.L. Honaker, Jr. and Patrick De Geest

5986 = 2 x 41 x 73

the digits of n and the prime factorization of n
are from the set 1 to 9 with each digit occurring exactly one time.

28651 = 7 x 4093

the digits of n and the prime factorization of n
are from the pandigital set 0 to 9 with each digit occurring exactly one time.

14368485 = 3 x 5 x 17 x 29 x 29 x 67

the digits of n and the prime factorization of n
are from the set 1 to 9 with each digit occurring exactly two times.

40578660 = 2 x 2 x 3 x 3 x 5 x 17 x 89 x 149

the digits of n and the prime factorization of n
are from the pandigital set 0 to 9 with each digit occurring exactly two times.

It has me wondering if such a composite number exists
in which each digit appears thrice and only thrice !

Special thanks goes to G. L. Honaker, Jr. for providing the original idea !
Prime Curios! sources : 5986 28651 14368485 40578660

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  Topic 3.4   [ August 13, 2000 ]
Friedman numbers

Problem of the month (August 2000) from Erich Friedman is very interesting
It investigates positive integers which can be written in some non-trivial way using its own digits,
together with the symbols + - x / ^ ( ) and concatenation.
The following subquestion is posed :

You might have fun confirming that 123456789 and 987654321 are Friedman numbers.

Solutions from Mike Reid and Philippe Fondanaiche

123456789 = ((86 + 2 * 7)⁵ – 91) / 3⁴

     and

987654321 = (8 * (97 + 6/2)⁵ + 1) / 3⁴

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  Topic 3.3  
Amazing Number Facts

Amazing Number Facts : Nine Digits 17469 divided by 5823 is 3. Every digit except 0 appears once and only once. [Kordemsky] 13485 divided by 2697 is 5. Idem. The prime 3187 divides 25496. Note all nine digits are used. 394521678 The sum of the prime numbers from 2 up to 92857 is a nine digits number. [Honaker] PI(2543568463) = 123456789. 5897230146 The pandigital sum of the first 32423 prime numbers. [Honaker] 12 * 34567 + 89 is prime. [Honaker]

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  Topic 3.2  
Difficult Digits Puzzle

Here is a pandigital puzzle I found at address http://pegasus.cc.ucf.edu/~mathed/digits.html :
Arrange the digits 0 1 2 3 4 5 6 7 8 9 using only one operation sign
so that they will equal 100. Good Luck!
As I don't know the answer(s) myself yet I'd appreciate if someone could send them to me!

On [ Januari 20, 2001 ] I received the following solution from Eric Poindessault (email)
He used as asked only operation sign namely the minus sign ¬
1 – – 2 – – 3 – – 4 – – 5 – 6 – 7 – – 8 – – 90 = 100
Thanks Eric, well done!

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  Topic 3.1  
The Nine Digits staircase

Gérard Villemin maintains an exciting maths website (in the French language)
and some of the pages deals also with Nine Digits and Pandigital numbers :

Nombres Pandigitaux

Arrange the nine digits in steps and rises (paliers et marches [Fr.]) of length 3
such that the sums are always... 13 !

9	3	1		13
		8
13		4	7	2
				5
		13		6

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For reference goals and easy searching I list here all the nine- & pandigitals implicitly displayed in these topics.

Topic 3.6 → 912768534
304678952, 348716902, 428563709, 407528936, 428507963, 406739852, 436718902, 502614798, 538604719, 513604978, 509763824, 536704981

Topic 3.5 → 598624173
2865174093

Topic 3.4 → 862759134, 897625134

Topic 3.3 → 174695823, 134852697, 318725496, 123456789

Topic 3.2 → 1234567890

Topic 3.1 → 931847256

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Contributions

Jean Claude Rosa (email) - go to topic

Terry Trotter (email) - go to topic

Shyam Sunder Gupta (email) - go to topic

Eric Pointdessault (email) - go to topic

Paul E. Triulzi (email) - go to topic

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(© All rights reserved ) - Last modified : October 2, 2023.

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Patrick De Geest - Belgium - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

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