7744

7744 is a positive even integer and perfect square, specifically $88^2$882, with prime factorization $2^6 \times 11^2$26×112.^[1][2] It is notable in recreational mathematics as the smallest known square number whose decimal digits form no isolated singles—all digits appear in adjacent pairs or groups.^[3] It is a Harshad number in base 10 and can be expressed as the sum of two fifth powers: $7^5 + (-2)^5 = 7744$75+(−2)5=7744. Additionally, 7744 is the side length (in units) of the smallest square that can be tiled with 21 smaller squares of distinct sizes, highlighting its combinatorial significance.^[3]

Mathematical Properties

• Divisibility and Factors: As an even number divisible by 2, 4, 8, 16, 32, and 64, 7744 has 21 positive divisors, including 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 121, 176, 242, 352, 484, 704, 968, 1372, 1936, and 7744 itself.^[2] The sum of its positive divisors is 16891 ($\sigma(7744) = 16891$σ(7744)=16891), so its abundance is 1403, classifying it as abundant.^[2]
• Digit and Numeric Curiosities: The digits of 7744 sum to 22 (digital root 4), and in binary, it is 1111001000000 (Hamming weight 5). It is not prime, odd, or a palindrome.^[1]
• Geometric Interpretations: 7744 relates to other geometric counts in combinatorial problems, such as squared square tilings, underscoring its utility in enumerative geometry.^[3]

This number appears in various mathematical contexts but lacks broader cultural or historical significance beyond these properties.

Mathematics

Square number properties

7744 is a perfect square, specifically equal to $88^2$882. This identity can be verified through direct computation using the binomial theorem: $88 = 80 + 8$88=80+8, so $$88^2 = (80 + 8)^2 = 80^2 + 2 \times 80 \times 8 + 8^2 = 6400 + 1280 + 64 = 7744.$$882=(80+8)2=802+2×80×8+82=6400+1280+64=7744. ^[4] As the square of the integer 88, 7744 occupies the position of the 88th perfect square in the sequence of squares of natural numbers. It lies between $87^2 = 7569$872=7569 and $89^2 = 7921$892=7921, illustrating its placement among consecutive squares.^[5][6] Since 88 is even, 7744 is an even perfect square, a property shared by squares of all even integers. The square root of 7744 can be computed using historical methods such as the digit-by-digit calculation algorithm, akin to long division, which originated in ancient Indian mathematics around the 5th century and was refined in medieval Europe. In this method, the digits of 7744 are grouped in pairs from the right (77 | 44). The largest square less than or equal to 77 is 64 ($8^2$82), subtracted to yield 13; bringing down 44 gives 1344. The current divisor is doubled to 16, and a digit $x$x is found such that $(160 + x) \times x \leq 1344$(160+x)×x≤1344; $x = 8$x=8 works exactly since $168 \times 8 = 1344$168×8=1344, confirming the root as 88 with no remainder.^[7] This approach provides an exact integer result for perfect squares like 7744, highlighting the efficiency of manual computation for such numbers before modern calculators.^[8]

Digit and numerical properties

7744 features a distinctive decimal representation consisting solely of the digits 7 and 4, each appearing exactly twice in adjacent pairs (77 and 44), with no isolated digits. This makes it the smallest known perfect square exhibiting this pattern of repeated adjacent digits without singles.^[3][9] In binary, 7744 is represented as 1111001000000₂, a 13-bit number with a Hamming weight of 5 (five 1s). The decimal-to-binary conversion involves successive division by 2 and recording remainders from least to most significant bit: starting with 7744 (even, remainder 0), divide to 3872 (even, 0), 1936 (even, 0), 968 (even, 0), 484 (even, 0), 242 (even, 0), 121 (odd, 1), 60 (even, 0), 30 (even, 0), 15 (odd, 1), 7 (odd, 1), 3 (odd, 1), 1 (odd, 1), yielding the binary string when remainders are reversed.^[2] 7744 is a Harshad (or Niven) number in bases 5, 9, 10, 12, 14, and 15, meaning it is divisible by the sum of its digits in each of those bases. For example, in base 10, the digits sum to 7 + 7 + 4 + 4 = 22, and 7744 ÷ 22 = 352, an integer.^[10][2] The sum of its decimal digits is 22, leading to a digital root of 4 via iterative summation (2 + 2 = 4). Additionally, 7744 is not a palindromic number, as its digits do not read the same forwards and backwards.^[2]

Algebraic identities

One notable algebraic representation of 7744 involves sums of fifth powers, specifically as the sum of two integer fifth powers:
$7744 = 6^5 + (-2)^5.$7744=65+(−2)5.
This identity holds because $6^5 = 7776$65=7776 and $(-2)^5 = -32$(−2)5=−32, so $7776 - 32 = 7744$7776−32=7744.^[11] Euler's totient function, denoted $\phi(n)$ϕ(n), counts the positive integers up to $n$n that are relatively prime to $n$n. For $n = 7744$n=7744, the value is $\phi(7744) = 3520$ϕ(7744)=3520. This result follows from the formula $\phi(n) = n \prod_{p \mid n} (1 - 1/p)$ϕ(n)=n∏p∣n​(1−1/p), where the product is over the distinct prime factors of $n$n, though the explicit prime factorization is addressed elsewhere.^[2]

Divisibility and factorization

The prime factorization of 7744 is $2^6 \times 11^2$26×112. This can be obtained through successive division by the smallest prime factors: $7744 \div 2 = 3872$7744÷2=3872, $3872 \div 2 = 1936$3872÷2=1936, $1936 \div 2 = 968$1936÷2=968, $968 \div 2 = 484$968÷2=484, $484 \div 2 = 242$484÷2=242, $242 \div 2 = 121$242÷2=121, and $121 = 11^2$121=112.^[12][13] The positive divisors of 7744, derived from its prime factorization, are: 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 121, 176, 242, 352, 484, 704, 968, 1936, 3872, and 7744.^[13][2] As an even number ending in 44, 7744 is divisible by 2 and by 4 (since the last two digits form 44, and $44 \div 4 = 11$44÷4=11). It is also divisible by 8 (last three digits 744, and $744 \div 8 = 93$744÷8=93) and by 11 (alternating sum of digits: $7 - 7 + 4 - 4 = 0$7−7+4−4=0).^[2] (Note: Standard divisibility rules from mathematical references; specific application verified via factorization.) The sum of the divisors function $\sigma(7744) = 16891$σ(7744)=16891, so the aliquot sum (sum of proper divisors) is $16891 - 7744 = 9147$16891−7744=9147.^[2][14] 7744 itself serves as the aliquot sum for the numbers 10316 and 15482, meaning the sum of proper divisors of each is 7744. For 10316, whose prime factorization is $2^2 \times 2579$22×2579, the proper divisors sum to 7744. For 15482 = $2 \times 7741$2×7741 (with 7741 prime), the proper divisors are 1, 2, and 7741, summing to 7744.^[15][16][17]

In popular culture

LEGO City set

LEGO set 7744, titled Police Headquarters, was released in 2008 as part of the LEGO City theme, designed by Raphael Pretesacque.^[18] It consists of 953 pieces and includes 7 minifigures: five police officers (with variations such as a sergeant with beard and glasses, a pilot with reddish brown hair, and an officer with a leather jacket) and two prisoners in striped outfits.^[19] The set is recommended for ages 6-12 and features a multi-level police headquarters building with a jail cell that has opening bars, a garage for a police vehicle with opening doors, a rooftop helicopter pad, and various accessories including computers, handcuffs, walkie-talkies, wanted posters, and a police dog.^[20][18] At launch, the set had a retail price of £69.99 in the UK and $89.99 in the US.^[18] On the secondary market, new sealed sets are valued at around $336 as of 2024, reflecting its popularity among collectors.^[18] Building instructions are available for download from the official LEGO website.^[20]

Other media references

Several educational YouTube videos focus on calculating the square root of 7744, primarily using step-by-step prime factorization and long division methods to teach mathematical concepts to students. For instance, a tutorial uploaded in November 2020 explains the process in detail for class 8 mathematics curricula.^[21] Another video from 2021 demonstrates the prime factorization approach, emphasizing its application in square root extraction.^[22] In number theory discussions, 7744 appears in Erich Friedman's online list "What's Special About This Number?", where it is highlighted as the smallest known square with no isolated digits, underscoring its unique digit properties.^[3] This reference contributes to its minor recognition in recreational mathematics puzzles and curiosities.

Other uses

7744 Ranch

The 7744 Ranch is a luxury adults-only resort located in East Austin, Texas, at 7744 Texas Highway 130, providing a serene natural escape just 15 minutes from downtown Austin, Austin-Bergstrom International Airport, the Circuit of the Americas, and the Tesla Gigafactory.^[23] This property offers private estates designed for high-end getaways, retreats, and events, blending modern luxury with the tranquility of a rural setting while remaining conveniently accessible to the city's vibrant scene.^[23] Established as a premium ranch experience, the 7744 Ranch features five expansive private estates, each spanning 1,200 square feet—four times the size of a typical hotel room—and equipped with full kitchens, private patios, and upscale furnishings inspired by opulent designs such as yacht aesthetics.^[24] Guests can book individual estates or the entire property for groups, accommodating romantic escapes, weddings, business retreats, or personal rejuvenation, with an emphasis on privacy and restoration in a nature-rich environment.^[23] Shared amenities enhance the luxury appeal, including a wood-fired oven for artisanal pizza-making, a koi pond for evening relaxation, an outdoor TV setup for stargazing entertainment, and wellness facilities like a sauna and hot tub.^[23] Outdoor activities at the ranch prioritize peaceful immersion in nature, with features such as a cowboy-style plunge pool for cooling off, well-maintained trails for leisurely walks, and fire pits for gatherings under the Texas sky.^[25] The property's design supports capacities for both intimate couples and larger groups, with bookings available directly through its online platform for flexible stays tailored to events or quiet weekends away.^[23] Overall, the 7744 Ranch delivers a high-end hospitality experience rated 4.9 out of 5 by guests for its resort-like ambiance and seamless integration of comfort and escape.^[26]

Color specifications

Pantone 7744 C is a vibrant green-yellow hue from the Solid Coated library, primarily utilized in professional graphics, printing, and design applications for its bold, earthy tone.^[27] This color, often approximated in digital formats with the hex code #BABC16 and RGB values (186, 188, 22), evokes a sense of natural warmth and is suitable for branding, packaging, and visual communications.^[28] Color harmony suggestions include analogous pairings with Pantone 368 C and 153 C for cohesive schemes, complementary contrasts with 2736 C for visual impact, and monochromatic variations with shades like 601 C and tints toward Black 3 C to maintain unity in designs.^[27] In contrast, Sherwin-Williams SW 7744, named Zeus, represents a sophisticated greige—a blend of gray and beige—with subtle green undertones, positioned as an ageless neutral ideal for interior painting projects.^[29] Described as bringing grounded, natural elegance to spaces, it works well on walls, trim, and cabinetry in settings like bedrooms, living rooms, kitchens, and bathrooms, offering a timeless backdrop that enhances refined aesthetics without overwhelming the environment.^[29] Its digital approximation is hex #99907E with RGB values (153, 144, 126), reflecting a muted, earthy sophistication suitable for modern and traditional interiors.^[30] These color specifications differ fundamentally in purpose and medium: Pantone 7744 C targets precise reproduction in print and digital graphics with its saturated, yellow-leaning vibrancy, while Sherwin-Williams Zeus focuses on practical paint applications, delivering a softer, versatile neutrality with green-infused depth for architectural use.^[27][29] Both embody an earthy elegance, but the former energizes visual designs, whereas the latter grounds physical spaces in subtle harmony.