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The s–m–n theorem states that if φ e {\displaystyle \varphi _{e}} is an ( m + n ) {\displaystyle (m+n)} place computable partial function and a 1 , … , a m {\displaystyle a_{1},\ldots ,a_{m}} are natural numbers, then there exists a primitive recursive function s n m {\displaystyle s_{n}^{m}} such that φ s n m ( e , a 1 , … , a m ) ≃ λ y 1 ⋯ y n [ φ e ( x 1 , … , x m , y 1 , … , y n ) ] {\displaystyle \varphi _{s_{n}^{m}(e,a_{1},\ldots ,a_{m})}\simeq \lambda y_{1}\cdots y_{n}[\varphi _{e}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})]} .
is an 
place computable partial function and 
are natural numbers, then there exists a primitive recursive function 
such that ![{\displaystyle \varphi _{s_{n}^{m}(e,a_{1},\ldots ,a_{m})}\simeq \lambda y_{1}\cdots y_{n}[\varphi _{e}(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da691513388a92532626fe4d68fc835a2f6bd5c3)
.