Problem of the Month (April 2012)

Friedman numbers, numbers that can be written in a non-trivial way using their own digits using only +, –, ×, /, ^, concatenation of digits, and any number of parentheses, was the August 2000 Math Magic problem. There are several other web sites allowing differing operations: Vampire Numbers, Narcissistic Numbers, Wild Narcissistic Numbers, Pretty Wild Narcissistic Numbers. This month we investigate a related question with what we will generally call self-reproducing numbers. Which numbers can be written using only one or two operations? We allow !, √, and decimal points as well, since they are popular choices among recreational math enthusiasts. What about Fibonacci numbers, Catalan numbers, or binomial coefficients? What other operations yield self-reproducing numbers?

ANSWERS

Zsbán Ambrus, Mark Mammel, Joe DeVincentis, and Bryce Herdt sent solutions.

As before, numbers are nicer if they can written with the digits in the original order, and below these are shown in red.

Exponentiation

25 = 5²	4096 = (4⁶)^9⁰ (MM)	15625 = (5⁶)^{1^2⁵}	19683 = (3⁹)^{1^6⁸}	59049 = (9⁵)^{4^0⁹}
78125 = (5⁷)^{1^8²}	117649 = ((7⁶)^1⁴)^1⁹	390625 = (5^2³)^{6^0⁹} (MM)	1953125 = ((5⁹)^1³)^{1^2⁵}	5764801 = ((7⁸)^{1^5⁶})^4⁰
10077696 = (((6⁹)^1⁶)^7⁰)^7⁰	40353607 = (((7³)³)^4⁰)^{5^0⁶}	43046721 = (((3⁴)⁴)^6⁰)^{1^2⁷}	134217728 = ((8^3²)^{1^4⁷})^{1^2⁷} (JD)	244140625 = (((5²)⁶)^{1^4⁴})^{2^0⁴}
387420489 = ((3²)⁹)^{8^{0^{7^{4^4⁸}}}}

Exponentiation and Addition

125 = 5¹⁺²	216 = 6²⁺¹	289 = (8 + 9)²	343 = (3 + 4)³	347 = 7³ + 4
736 = 7 + 3⁶	2187 = (2 + 1⁸)⁷	3125 = (3¹ + 2)⁵	3375 = (3 + 5 + 7)³	3378 = (7 + 8)³ + 3
13824 = (8¹ + 2⁴)³	14641 = (1 + 4 + 6)^4¹	14645 = (5 + 6)⁴ + 4¹	14647 = (4 + 7)⁴ + 6¹	15626 = 5⁶ + 1^2⁶
15627 = (5⁶)^1⁷ + 2	15628 = 5⁶ + 1⁸ + 2	15631 = (5¹⁺¹)³ + 6	15632 = (5²)³ + 6 + 1	15641 = 5⁶ + 4¹⁺¹
15642 = 1 + 5⁶ + 4²	15661 = 5⁶ + 6¹⁺¹	15662 = 1 + 5⁶ + 6²	16384 = 4^{6+1^3⁸}	16815 = (1 + 6)⁵ + 8¹
17536 = 1 + 7⁵ + 3⁶	19732 = 3⁹ + 7^1²	32771 = 3 + 2⁷⁺⁷⁺¹	32775 = 2³⁺⁵⁺⁷ + 7	32778 = 2⁷⁺⁸ + 7 + 3
38416 = (6^1³ + 8)⁴	44977 = (7 + 7)⁴ + 9⁴	46660 = (4 + 6⁶)^6⁰	46661 = 6⁶ + 4 + 1⁶	59050 = 9⁵ + 5⁰ + 0
59051 = 9⁵ + 5⁰ + 1	59052 = 9⁵ + 5⁰ + 2	59053 = 9⁵ + 5⁰ + 3	59054 = 9⁵ + 5⁰ + 4	59055 = 9⁵ + 5⁰ + 5
59056 = 9⁵ + 5⁰ + 6	59057 = 9⁵ + 5⁰ + 7	59058 = 9⁵ + 5⁰ + 8	59059 = 9⁵ + 5⁰ + 9	78225 = 5⁷ + (2 + 8)²
104976 = (1 + 9 + 7 + 6⁰)⁴ (JD)	117650 = 1¹ + (7⁶)^5⁰	1679616 = 6^{(7+1)^{1^{9^6⁶}}} (JD)	2097152 = (2⁷)^{2+1+0⁹⁺⁵} (JD)	2097153 = 2^0+7+9+5 + 1³ (JD)
16777216 = (1⁶⁺⁷⁺⁷ + 7)^2¹+6 (JD)

Exponentiation and Subtraction

127 = 2⁷ – 1	128 = 2^8–1	625 = 5^6–2	4088 = 8⁴ – 8 – 0
13825 = 1 – (8 – 2⁵)³	15617 = 5⁶ – 7¹ – 1	15618 = 1 – (8¹ – 5⁶)	15621 = 5⁶ – 2 – 1 – 1
15622 = 5⁶ – 2 – (2 – 1)	15623 = 5⁶ – (3 – 2) – 1	15624 = 5⁶ – (4 – 2 – 1)	15626 = 5⁶ – (1⁶ – 2)
15632 = 5⁶ – (1 – 2³)	15633 = 5⁶ – (1 – 3)³	16374 = 4⁷ – 1 – 6 – 3	16375 = (5 – 1)⁷ – 6 – 3
16377 = (1 – (3 – 6))⁷ – 7	16378 = (8 – 3 – 1)⁷ – 6	16384 = 4^{8–1^6–3}	16743 = 7^6–1 – 4³
16794 = 7^6–1 – 9 – 4	16798 = 7^6–1⁸ – 9	16807 = 7^6–1 – 0⁸	19692 = 9 – (1 – (2 – 6))⁹
19693 = 1 – ((3 – 6)⁹ – 9)	32759 = (3 – (2 – 7))⁵ – 9	32761 = (2^6–1)³ – 7	32762 = (2^7–2)³ – 6
32768 = ((8 – 6)³)^7–2	32775 = 7 – (2 – 3 – 7)⁵	43264 = (6³ – 4 – 4)²	45632 = (6³)² – 4⁵
46633 = 6⁶ – (3³ – 4)	46644 = 6⁶ – 4 – 4 – 4	46648 = (4 – (4 – 6))⁶ – 8	46650 = 6⁶ – 5 – 4⁰
46651 = 6⁶ – 1⁵ – 4	46653 = 6⁶ – 3^5–4	46655 = 6⁶ – 4^5–5	46656 = (6⁶)^(5–4)⁶
46657 = 6⁶ – (4 – 5)⁷	46660 = 4 – (0⁶ – 6⁶)	46661 = 6⁶ – (1⁴ – 6)	46663 = 6⁶ – (3 – 4 – 6)
54958 = 9⁵ – (8⁴ – 5)	58921 = 9⁵ – 2^8–1	59038 = 9⁵ – 8 – 3 – 0	59039 = 9⁵ – 9 – 3⁰
59044 = 9⁵ – 4 – 4⁰	59045 = 9⁵ – 4 – 0⁵	59046 = 9⁵ – (4 – 6⁰)	59048 = 9⁵ – (4⁰)⁸
59050 = 9⁵ – (0 – 5⁰)	59052 = 9⁵ – 2 – (0 – 5)	59053 = 9⁵ – (3⁰ – 5)	59054 = 9⁵ – (0⁴ – 5)
59129 = 9⁵ – (1 – 9²)	59263 = 9⁵ – (2 – 6³)	59392 = 9⁵ – (2 – 9)³	78115 = 5⁷ – 8 – 1 – 1
78116 = (6 – 1)⁷ – 8 – 1	78117 = (6 – 1)⁷ – 8¹	78123 = (8 – 3)⁷ – 2¹	78133 = 8 – (1 – 3 – 3)⁷
78135 = 8 – (1 – 5⁷ – 3)	78136 = 8 – ((1 – 6)⁷ – 3)	97966 = 7⁶ – (9 – 6)⁹	246519 = 4⁹ – 5^{6^1²} (JD)
248832 = (8–(2–2–4))^8–3 (JD)	259743 = 4⁹ – 7^5–(3–2) (JD)	278640 = 6⁷ – (8 – 2)⁴ – 0 (JD)	279680 = 6⁷ – 2⁸ – 0⁹ (JD)
279936 = 6⁷ – (9 – 9)^3/2 (JD)	524284 = 2^{8–(2–5–8)} – 4 (JD)	524288 = 2^{8–(4–5–2–8)} (JD)	531441 = 3^{(5¹–(1–4–4))} (JD)
1048576 = 4^{5–(8–7–6)–0¹} (JD)	1679608 = 6⁸ – 7 – 1^{9^6⁰} (JD)	1679616 = 6^{(9–1)^{1^{7^6⁶}}} (JD)	1946564 = 5⁹ – (4–(1⁶–6))⁴ (JD)
1950724 = 5⁹ – 7⁴ – 0^1² (JD)	1952395 = 5⁹ – 9³ – 1^2⁵ (JD)	1952396 = 5⁹ – 9^{3^{6^1²}} (JD)	1952868 = 5⁹ – 2⁸ – 1^6⁸ (JD)
1952869 = 5⁹ – 2^{8^{1^6⁹}} (JD)	1952997 = 5⁹ – 2^{7^{1^9⁹}} (JD)	1953042 = 5⁹ – 3⁴ – 2^1⁰ (JD)	1953043 = 5⁹ – 3⁴ – 1^3⁰ (JD)
1953044 = 5⁹ – 3⁴ – 0^1⁴ (JD)	1953113 = (5³)³ – 9 – 1 – 1 – 1 (JD)	1953114 = (5³)^4–1 – 9 – 1 – 1 (JD)	1953115 = 5⁹ – 5 – 3 – 1 – 1¹ (JD)
1953116 = 5⁹ – 6 – 3^{1^1¹} (JD)	1953117 = 5⁹ – 7 – 1^{1^1³} (JD)	1953118 = 5⁹ – (8–1)^{1^1³} (JD)	1953119 = 5⁹ – 3 – 1 – 1 – 1⁹ (JD)
1953120 = 5⁹ – 3¹ – 2^1⁰ (JD)	1953121 = 5⁹ – 3^{1^2¹} – 1 (JD)	1953122 = 5⁹ – 3^{1^{2^2¹}} (JD)	1953123 = 5⁹ – 2^{1^{1^3³}} (JD)
1953124 = 5⁹ – 1^{3^{1^2⁴}} (JD)	2097151 = 2^{(9–(0¹–7–5))} – 1 (JD)	2097152 = 2^{(9–(0^2–1–7–5))} (JD)	4194300 = 4^{9–(1–3–0)} – (4 – 0) (JD)
4194301 = 4^{9–(1–3–0)} – (4 – 1) (JD)	4194302 = 4^{9–(1–3–0)} – (4 – 2) (JD)	4194303 = 4^{9–(1–3–0)} – (4 – 3) (JD)	4194304 = 4^{9–(1–3–0)} – (4 – 4) (JD)
4194305 = 4^{9–(1–3–0)} – (4 – 5) (JD)	4194306 = 4^{9–(1–3–0)} – (4 – 6) (JD)	4194307 = 4^{9–(1–3–0)} – (4 – 7) (JD)	4194308 = 4^{9–(1–3–0)} – (4 – 8) (JD)
4194309 = 4^{9–(1–3–0)} – (4 – 9) (JD)	4782969 = 9^{7^{(9–8)^{2^4⁶}}} (JD)	8388608 = (8 – 6)^{8–(3⁰–8–8)} (JD)

Exponentiation and Multiplication

2592 = 2⁵ × 9²	2916 = (1 × 6 × 9)²	23328 = (2 × 3³)² × 8	28672 = 7 × 2⁶ × 8²
32805 = 5 × 3⁸ × 2⁰	34805 = 5^2⁰ × 3⁸	34992 = 3 × 9 × (4 × 9)²	82944 = (8 × 9)² × 4 × 4
262144 = (2 × 4)⁶ × 1^2×4 (JD)	1594323 = 9⁴ × 3⁵ × 1^2×3 (BH)	1679616 = 6⁷ × 6¹ × 1^6×9 (JD)	14348907 = 3⁷ × 9⁴ × 1^4×8×0 (BH)

Exponentiation and Division

11664 = 1 / (1 / 6⁶)) / 4 (BH)	46656 = 6^(6⁵/6⁴) (JD)	279936 = 6^{7^{(9/9)^3/2}} (JD)	2097152 = (2⁷)^{9^{1/2^5⁰}} (JD)	1679616 = 6⁹ / 6¹ / 1^6/7 (JD)
4782969 = 9^{7^{(8/4/2)^6⁹}} (JD)

Exponentiation and Concatenation

121 = 11²	9216 = (96²)¹	11025 = (105²)¹	13824 = (24³)^1⁸	19321 = (139²)¹
39304 = (34³)^9⁰	81225 = (285²)¹	131072 = 2^{17^1³⁰} (JD)	161051 = 11^{5^16⁰} (BH)	1771561 = 11^{6^1⁵⁷⁷} (BH)
2097152 = 2^{21^975⁰} (JD)	19487171 = 11^{7^{1⁴⁷⁸⁹}} (BH)

Exponentiation and Decimal Points

2097152 = (2⁷)^{9^{0.5^1²}} (JD)

Roots

0 = √(0)

1 = √(1)

Roots and Exponentiation

64 = √(4)⁶

2097152 = (2⁷)^{√9^{1^{2^5⁰}}}(JD)

Roots and Decimal Points

1024 = ¹√(^0.2√4) (JD)

3125 = ^(³√1)√(^.2√(5)) (JD)

Factorial

1 = 1!

2 = 2!

Factorial and Addition

144 = (1 + 4)! + 4!	145 = 1 + 4! + 5!	733 = 7 + 3! + (3!)!	5160 = 5! + (1 + 6)! + 0
5161 = 5! + (1 + 6)! + 1	5162 = 5! + (1 + 6)! + 2	5163 = 5! + (1 + 6)! + 3	5164 = 5! + (1 + 6)! + 4
5165 = 5! + (1 + 6)! + 5	5166 = 5! + (1 + 6)! + 6	5167 = 5! + (1 + 6)! + 7	5168 = 5! + (1 + 6)! + 8
5169 = 5! + (1 + 6)! + 9	40440 = (4 + 0!)! + (4 + 4)! + 0	40441 = (4 + 0!)! + (4 + 4)! + 1	40442 = (4 + 0!)! + (4 + 4)! + 2
40443 = (4 + 0!)! + (4 + 4)! + 3	40444 = (4 + 0!)! + (4 + 4)! + 4	40445 = (4 + 0!)! + (4 + 4)! + 5	40446 = (4 + 0!)! + (4 + 4)! + 6
40447 = (4 + 0!)! + (4 + 4)! + 7	40448 = (4 + 0!)! + (4 + 4)! + 8	40449 = (4 + 0!)! + (4 + 4)! + 9	40584 = (4 + 0!)! + 5! + 8! + 4!
40585 = 4! + 0! + 5! + 8! + 5!	363000 = (3 + 6)! + (3 + 0! + 0!)! + 0 (BH)	363001 = (3 + 6)! + (3 + 0! + 0!)! + 1 (BH)	363002 = (3 + 6)! + (3 + 0! + 0!)! + 2 (BH)
363003 = (3 + 6)! + (3 + 0! + 0!)! + 3 (BH)	363004 = (3 + 6)! + (3 + 0! + 0!)! + 4 (BH)	363005 = (3 + 6)! + (3 + 0! + 0!)! + 5 (BH)	363006 = (3 + 6)! + (3 + 0! + 0!)! + 6 (BH)
363007 = (3 + 6)! + (3 + 0! + 0!)! + 7 (BH)	363008 = (3 + 6)! + (3 + 0! + 0!)! + 8 (BH)	363009 = (3 + 6)! + (3 + 0! + 0!)! + 9 (BH)	363024 = (3 + 6)! + (3 + 0 + 2)! + 4! (BH)
363613 = (3 + 6)! + 3! + 6! + 1 + 3! (BH)

Factorial and Subtraction

715 = (7 – 1)! – 5	720 = (7 – (2 – 0!))!	1435 = (4 – 1)!! – (5 – 3!!)	1463 = 4! – (1 – 6! – 3!!)
4296 = (9 – 2)! – 4! – 6!	4317 = 7! – 3!! – (4! – 1)	5017 = 7! – ((5 – 1)! – 0!)	5035 = ( 5 – (0! – 3))! – 5
5037 = 7! – 0! – (5 – 3)	5038 = (8 – 0)! – (5 – 3)	5039 = (9 – (5 – 3))! – 0!	5040 = (5 – (0! – 4) – 0!)!
5637 = 7! – (5! – 6!) – 3	6476 = 6! – (4 – 7! – 6!)	10077 = -1 – 0! – (0! – 7! – 7!)	10078 = (8 – 0!)! – 0! – (1 – 7!)
10079 = (9 – 0! – 0!)! – (1 – 7!)	35280 = 8! – (5 – (2 – 3 – 0!))!	39600 = (9 – (3 – 0! – 0!))! – 6!	40287 = 8! – 4! – 7 – 2 – 0
40289 = 8! – 4! – (9 – 2 – 0)	40294 = (9 – 0!)! – 4! – (4 – 2))	40296 = (9 – 0!)! – (6 – (4 – 2))!	40298 = 8! – (9 – 4 – 0!)! – 2
40308 = 8! – 4 – 0! – 3! – 0!	40309 = (9 – 0!)! – 4 – 0! – 3!	40314 = (4 – (1 – 0! – 4))! – 3!	40315 = (5 – (0 – 3))! – 4 – 1
40316 = (6 – (1 – 3))! – 4 – 0	40318 = 4 – 0! – (3! – 1 – 8!)	40319 = (9 – 1)! – (4 – 3 – 0)	40320 = (4 – (0! – 3 – 2))! – 0
40321 = (3! – (2 – 4))! – (0 – 1)	40322 = (3! – (2 – 4))! – (0 – 2)	40323 = (3! – (2 – 4))! – (0 – 3)	40324 = (3! – (2 – 4))! – (0 – 4)
40325 = (3! – (2 – 4))! – (0 – 5)	40326 = (3! – (2 – 4))! – (0 – 6)	40327 = (3! – (2 – 4))! – (0 – 7)	40328 = 4 – (0! – 3 – 2 – 8!)
40329 = (3! – (2 – 4))! – (0 – 9)

Factorial and Multiplication

36 = 3! × 6

40320 = (4 × (0 × 3)! × 2)! × 0! (JD)

82944 = 8 × 2 × 9 × 4! × 4!

Factorial and Division

120960 = 9! / (6/2) / 1 / 0! / 0! (JD)	255024 = 4!! / (5 / ((0/5)! / 2 / 2))!(JD)
362880 = (3 / (2/6))! / (8/8) / 0! (JD)	479001600 = (9! / 7! / 6)! / (0/4)! / 0! / 0! / 0! (JD)

Factorial and Exponentiation

13824 = (4!)^{3^{1^8²}} (JD)

Multiplication and Concatenation

126 = 6 × 21	153 = 3 × 51	688 = 8 × 86	1206 = 6 × 201	1255 = 5 × 251
1260 = 6 × 210	1395 = 15 × 93	1435 = 35 × 41	1503 = 3 × 501	1530 = 3 × 510
1827 = 21 × 87	2187 = 27 × 81	3159 = 9 × 351	3784 = 8 × 473	6880 = 8 × 860
10251 = 51 × 201	10255 = 5 × 2051	10426 = 26 × 401	10521 = 21 × 501	10525 = 5 × 2105
10575 = 15 × 705	11259 = 9 × 1251	11439 = 9 × 31 × 41	11844 = 84 × 141	11848 = 8 × 1481
12006 = 6 × 2001	12060 = 6 × 2010	12384 = 3 × 4128	12505 = 5 × 2501	12546 = 51 × 246
12550 = 5 × 2510	12595 = 5 × 2519	12600 = 6 × 2100	12762 = 6 × 2127	12768 = 8 × 21 × 76
12798 = 2 × 79 × 81	12843 = 3 × 4281	12955 = 5 × 2591	12964 = 14 × 926	13243 = 41 × 323
13545 = 3 × 4515	13764 = 6 × 31 × 74	13950 = 15 × 930	14035 = 35 × 401	14350 = 35 × 410
15003 = 3 × 5001	15030 = 3 × 5010	15246 = 6 × 2541	15300 = 3 × 5100	15435 = 3 × 5145
15624 = 24 × 651	15795 = 9 × 1755	15975 = 5 × 5 × 9 × 71	16272 = 6 × 2712	16368 = 8 × 31 × 66
17325 = 75 × 231	17346 = 6 × 7 × 413	17428 = 2 × 8714	17437 = 47 × 371	17482 = 2 × 8741
17892 = 9 × 28 × 71	18225 = 81 × 225	18265 = 65 × 281	18270 = 21 × 870	19026 = 21 × 906
19215 = 21 × 915	19845 = 5 × 49 × 81	21375 = 3 × 7125	21586 = 86 × 251	21753 = 3 × 7251
21870 = 27 × 810	24768 = 4 × 72 × 86	25105 = 5 × 5021	25375 = 35 × 725	25474 = 47 × 542
25510 = 5 × 5102	28476 = 42 × 678	29184 = 4 × 8 × 912	29632 = 32 × 926	31509 = 9 × 3501
31590 = 9 × 3510	33579 = 7 × 9 × 533	33655 = 53 × 635	33696 = 36 × 936	34992 = 9 × 9 × 432
36855 = 63 × 585	37668 = 6 × 73 × 86	37840 = 8 × 4730	37845 = 87 × 435	39784 = 8 × 4973
41665 = 65 × 641	42898 = 89 × 482	44676 = 6 × 7446	45684 = 54 × 846	45760 = 65 × 704
45864 = 84 × 546	47538 = 57 × 834	48672 = 78 × 624	49855 = 59 × 845	49896 = 6 × 84 × 99
49968 = 8 × 9 × 694	51759 = 9 × 5751	52168 = 8 × 6521	53865 = 63 × 855	56295 = 9 × 6255
56875 = 65 × 875	62968 = 68 × 926	63895 = 65 × 983	67149 = 9 × 7461	67392 = 72 × 936
67950 = 75 × 906	68800 = 8 × 8600	69984 = 9 × 9 × 864	71199 = 9 × 7911	78975 = 9 × 8775

Fibonacci

0 = F₀

1 = F₁

5 = F₅

Fibonacci and Addition

55 = F₅₊₅	233 = F_{F_(F₂+3+3)}	234 = F₂ + F_F₃₊₄	235 = 2 + F_{F_(F₃+4)}	236 = 3 + F_{F_(F₂+6)}
237 = 2 + F₃ + F_F₇	267 = F_(F₂+F₆) + F_F₇	2584 = F_(F₂+5+8+4)	2585 = F₂ + F₅₊₈₊₅	2589 = F_(F₂+8+9) + 5
6772 = 6 + F_(F₇+7) + F₂	6773 = 6 + F_(F₇+7) + F₃	6774 = 6 + F_(F₇+7) + F₄

Fibonacci and Subtraction

376 = F_{(7–F_F₆)} – F_F₃	377 = F_{(F_{F_{(7–F_F₃)}}–7)}	1576 = F_{(F₇–(1–5))} – F_F₆	1596 = F_{(F_F₆–(9–5))} – 1
1597 = F_{(F₉–F₇–(5–1))}	2582 = F_{(2–(5–F₈))} – 2	2583 = F_{(F₃–(5–F₈))} – F₂	2584 = F_{(F₈–2–(5–4))}
4177 = F_{(F₇–(1–7))} – 4	4180 = F_{(F₈–F_F₄)} – 1 – 0	4181 = F_{(4–(1–F₈–1))}	6763 = F_{(F_F₆–(7–6))} – F₃ (BH)
6764 = F_{(F_F₆–(7–6))} – F_{F_F₄} (BH)	6765 = F_{(F_{(F_F₆–F₇)}–(6–5))}	10857 = F_F₈ – F_{0–(1–5–7)} (BH)	10943 = F_{F_{F_{1–(0–9)–4}}} – 3 (BH)
10944 = F_{F_{F_{1–(0–9)–4}}} – F_F₄ (BH)	10946 = F_{F_{1–(0–9–4)–6}} (BH)	17711 = F_(F₇–7) – (1–1–1) (BH)	28657 = F_{(2–(8–F_F₆)–(5–F₇))} (BH)

Fibonacci and Multiplication

34 = F_(3×F₄)	63 = F_F₆ × 3	84 = F₈ × 4	144 = 1 × F_(F₄×4)	168 = 1 × F₆ × F₈
189 = 1 × F₈ × 9	273 = F_{F_2×3} × F₇	495 = F_{(F_F₄×5)} × 9	882 = F₈ × F₈ × 2	1296 = 1 × F_2×6 × 9
2646 = 2 × F_F₆ × F₄ × F_F₆	2688 = 2 × F₆ × F₈ × 8

Lucas

1 = L₁

Lucas and Addition

32 = 3 + L_{L_{L_L₂}}	58 = L₅ + L₈	123 = L_{(1+2+L_L₃)}	124 = 1 + L_(L₂+L₄)	199 = L₁₊₉ + L₉
201 = 2 + L_{L_{(L_{L_L₀}+1)}}	202 = L₂ + L_{L_(L₀+L₂)}	203 = L_L₂ + L_{L_(L₀+3)}	205 = L_L₂ + L₀ + L_L₅	228 = L_{L_{L_L₂}} + L_(L₂+8)
235 = L_{L_L₂} + L_{L_L₃} + L_L₅	257 = L_{L_{L_L₂}} + L_L₅ + L₇	275 = L₂₊₇ + L_L₅	322 = L_{(3+2+L_{L_L₂})}	325 = 3 + L_{(L_{L_L₂}+5)}
329 = L_L₃ + L_(L₂+9)	351 = L_{L_L₃} + L_(L₅+1)	398 = L₉ + L_(L₃+8)	521 = L_{(5+L_{L_L₂}+1)}	526 = 5 + L_{(L_{L_L₂}+6)}
528 = L_{L_L₂} + L₅₊₈

Lucas and Subtraction

22 = L_{L_{L_L₂}} – L_{L_L₂}

27 = L₇ – 2

29 = L_{L_{L_{L_{(L_{L_L₂}–9)}}}}

123 = L_{(L_{(L_L₃–2)}–1)}

521 = L_{(L_{(L_{L_L₂}–1)}–5)}

843 = L_{(L₈–4–L_{L_L₃})}

Lucas and Multiplication

35 = L_L₃ × 5	55 = L₅ × 5	126 = 1 × L_{L_L₂} × L₆	145 = L_{L_1×4} × 5	203 = L_2×L₀ × L_{L_L₃}
232 = 2 × L₃ × L_{L_{L_L₂}}	235 = L_(2×L₃) × 5	245 = L_{L_L₂} × L₄ × 5	282 = 2 × L₈ × L₂	304 = L_{(3×L_L₀)} × 4
322 = L_3×2×2	343 = L_L₃ × L₄ × L_L₃	435 = L_L₄ × 3 × 5

Binomial

28 = ₈C₂ (ZA)

Binomial and Addition

6435 = _₆C₄C₃₊₅	6438 = _₆C₄C₈ + 3	18564 = _1+8+5+4C₆	18565 = ₈₊₅₊₅C₆ + 1	18569 = ₁₊₈₊₉C₆ + 5
20475 = _{_(7+₅C₀)C₂}C₄	26235 = _{₅₊₆C₂}C_₃C₂	75582 = _7+5+5+2C₈	75587 = ₇₊₅₊₇C₈ + 5	75589 = ₅₊₅₊₉C₈ + 7
92378 = ₃₊₇₊₉C₂₊₈	118755 = _{(₈C₁+₇C₅)}C_₅C₁	118756 = _{(1+_₈C₇C₆)}C₅ + 1	122265 = _{(₁₊₆₊₅C₂₊₂)}C₂	122266 = _{(₆₊₆C₂₊₂)}C₂ + 1
123753 = _{(₇₊₅C₁₊₃+3)}C₂	123754 = _{(₇₊₅C₄+3)}C₂ + 1	134596 = _{(₅C₃+1+4+9)}C₆	148995 = _{_{(1+₅C_₉C₉)}C₈}C₄	167960 = _1+6+7+6C₉ + 0
167961 = _1+6+7+6C₉ + 1	167962 = _1+6+7+6C₉ + 2	167963 = _1+6+7+6C₉ + 3	167964 = _1+6+7+6C₉ + 4	167965 = _1+6+7+6C₉ + 5
167966 = _1+6+7+6C₉ + 6	167967 = _1+6+7+6C₉ + 7	167968 = _1+6+7+6C₉ + 8	167969 = _1+6+7+6C₉ + 9	184756 = ₈₊₇₊₅C_(4+₆C₁)

Binomial and Subtraction

4996 = _₆C₄C₉ – 9	7139 = _₉C₇C₃ – 1	12376 = _{(₇C₂–1–3)}C₆	38760 = _₆C₃C_{(7–₈C₀)}	43758 = _{(₇C_₅C₄–3)}C₈
46376 = _{(₇C₃–₆C₆)}C₄	54738 = _₈C₄C₃ – (7 – 5)	77515 = _{(₇C₅–1)}C₇ – 5	77520 = _{(₇C₅–₂C₀)}C₇	92375 = _{(₇C₅–2)}C₉ – 3
92378 = _{(₇C_8–3–2)}C₉	95284 = _{₉C_₄C₂}C₃	98265 = _₈C₂C₅ – 6 – 9	116277 = _{₇C_6–1}C₇ – 1 – 2	116278 = _{_8–1C_6–1}C₇ – 2
116279 = _{_9–2C_6–1}C₇ – 1	125963 = _₆C₃C_9–1 – 2 – 5	125965 = _{₆C_5–2}C_9–1 – 5	125968 = _{₆C_8–5}C_9–1 – 2	125970 = _{(₇C₂–₅C₀)}C_9–1
134596 = _{(_9–1C_5–3–4)}C₆	167945 = _{(₇C₅–1)}C₉ – ₆C₄	167952 = _{(₇C₅–1)}C₉ – 6 – 2	167953 = _{₆C_5–(3–1)}C₉ – 7	167954 = _{₆C_7–4}C₉ – 1 – 5
167955 = _{₆C_7–(5–1)}C₉ – 5	167958 = _{(₇C₅–1)}C₉ – (8 – 6)	167959 = _{₆C_7–(9–5)}C₉ – 1	167960 = _{(₇C_6–1–₆C₀)}C₉	194580 = _{(₈C₅–9)}C₄ – ₁C₀
203487 = _₇C₂C₈ – ₃C_₄C₀

Binomial and Multiplication

2268 = 6 × _₈C₂C₂	2376 = 3 × _2×6C₇	15504 = _{(5×4×₁C₀)}C₅	17892 = 1 × 7 × _8×9C₂	22896 = 2 × 8 × _6×9C₂
24384 = 3 × _4×4×8C₂	24864 = 4 × _(4×₈C₆)C₂	32725 = 5 × _{(₇C_2×2)}C₃	36288 = 3 × 6 × _8×8C₂	39711 = 1 × 1 × _9×7C₃
74382 = 7 × _(8×₃C₂)C₄	82368 = 8 × _8×6×3C₂	92664 = 9 × _6×6×4C₂	123984 = 1 × 3 × _9×8×4C₂	139536 = 1 × 9 × _₆C₃C_3×5
142506 = ₁C₀ × _2×₆C₄C₅	174832 = 1 × 7 × _4×₈C₃C₂	198432 = 1 × 8 × _9×₄C₂C₃

Binomial and Fibonacci

56 = _F₆C₅ (ZA)

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